Integral geometry – measure theoretic

approach and stochastic applications

Rolf Schneider

PrefaceIntegral geometry, as it is understood here, deals with the computation andapplication of geometric mean values with respect to invariant measures.In the following, I want to give an introduction to the integral geometryof polyconvex sets (i.e., finite unions of compact convex sets) in Euclideanspaces. The invariant or Haar measures that occur will therefore be thoseon the groups of translations, rotations, or rigid motions of Euclidean space,and on the affine Grassmannians of k-dimensional affine subspaces. How-ever, it is also in a different sense that invariant measures will play a centralrole, namely in the form of finitely additive functions on polyconvex sets.Such functions have been called additive functionals or valuations in theliterature, and their motion invariant specializations, now called intrinsicvolumes, have played an essential role in Hadwiger’s [2] and later work (e.g.,[8]) on integral geometry. More recently, the importance of these functionalsfor integral geometry has been rediscovered by Rota [5] and Klain-Rota [4],who called them ‘measures’ and emphasized their role in certain parts ofgeometric probability. We will, more generally, deal with local versions ofthe intrinsic volumes, the curvature measures, and derive integral-geometricresults for them. This is the third aspect of the measure theoretic approachmentioned in the title. A particular feature of this approach is the essentialrole that uniqueness results for invariant measures play in the proofs.

As prerequisites, we assume some familiarity with basic facts from mea-sure and integration theory. We will also have to use some notions andresults from the geometry of convex bodies. These are intuitive and easy tograsp, and we will apply them without proof. In order to understand the ap-plications to stochastic geometry that we intend to explain, the knowledgeof fundamental notions from probability theory will be sufficient.

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The material is taken from different sources, essentially from the lecturenotes on “Integralgeometrie” [8] and “Stochastische Geometrie” [9], bothwritten together with Wolfgang Weil. Another source is the fourth chapterof the book [7] on convex bodies.

Contents

1 Introduction 3

2 Elementary mean value formulae 4

3 Invariant measures of Euclidean geometry 10

4 Additive functionals 24

5 Local parallel sets and curvature measures 29

6 Hadwiger’s characterization theorem 39

7 Kinematic and Crofton formulae 42

8 Extension to random sets 48

9 The kinematic formula for curvature measures 58

References 67

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1 Introduction

It will be one aim of the following lectures to develop some integral geometricformulae for sets in Euclidean space and to show how they can be appliedin parts of stochastic geometry. In particular, I want to emphasize the rolethat integral geometry can play in the theoretical foundations of stereology.By stereology one understands a collection of procedures which are usedto estimate certain parameters of real materials by means of measurementsin small probes and plane sections. Stereology is applied in biology andmedicine as well as in material sciences (e.g., metallography, mineralogy).

Since much of the motivation for the later theoretical investigationscomes from these practical procedures, let me first explain the underlyingideas by two typical examples.

In geology, one may be interested in determining the volume proportionof some mineral in a rock. Thus one assumes that for the material in ques-tion there is a well-defined parameter, traditionally denoted by VV , thatspecifies the volume of the investigated mineral per unit volume of the totalmaterial. In order to determine this specific volume VV , one will first takea probe of the material “at random”. As a second step, Delesse (1847) pro-posed to produce a (polished) plane section of the probe, possibly again “atrandom”, and to determine the specific area AA of the investigated mineralin that section. On the basis of heuristic arguments, Delesse asserted that

VV = AA,

or rather that the measured value AA is a good estimate for the unknownparameter VV .

A second example is taken from medicine. One may be interested in thegas exchange of a mammal lung, and this depends on the alveolar surfaceof the lung. To measure this specific area, denoted by SV , only a smallprobe of the lung tissue will be available, and usually only a thin slice canbe observed under the microscope. Tomkeieff (1945) proposed to determinethe specific boundary length LA of the tissue in the section and then toestimate the unknown specific area SV by means of the formula

SV =4πLA,

again supported by heuristic arguments.Scientists working in practice have developed similar formulae. The so-

called ‘fundamental equations of stereology’ are

VV = AA, SV =4πLA, MV = 2πχA.

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Here M denotes the integral of the mean curvature, and χ is the Eulercharacteristic.

It is evident that such heuristic procedures are implicitly based on manytacit assumptions. A theoretical justification has to begin by analyzing theseassumptions, it has to provide suitable models and must finally lead to ex-actly proven formulae of the type used in practice. The first assumption isthat the parameter of the material to be determined, like volume or surfacearea per unit volume, exists and can be estimated with sufficient accuracyfrom taking randomly placed probes and averaging. A solid foundation andjustification can be achieved if the material under investigation is modelledas the realization of a random set. Taking a probe at random can then bemodelled as follows. We fix a shape for the probe or ‘observation window’,say a compact convex set K with positive volume. Inside K we observe arealization Z(ω) of our random set Z. We assume that for the intersectionZ(ω) ∩K we are able to measure a geometric functional ϕ of interest, likevolume or surface area. Instead of placing K in a random position, oneassumes that the random set Z has a suitable invariance property, meaningthat Z and its image under any translation or rigid motion are stochasticallyequivalent. Under suitable model assumptions, the mathematical expecta-tion Eϕ(Z ∩ K) will exist, and the measured value ϕ(Z(ω) ∩ K) can beconsidered as an unbiased estimator. If the model is such that the randomset Z has a well-defined ϕ-density, the next question is then how this isrelated to the local expectation Eϕ(Z ∩ K), depending on the test bodyK. Similar considerations will be necessary to justify the determination ofparameters from randomly placed lower-dimensional sections.

This program, of which we have merely given a rough sketch, will obvi-ously require the development of

• a theory of random sets with suitable invariance properties, admittingdensities of geometric functionals, like volume, surface area, Eulercharacteristic,

• a theory of mean values of geometric functionals, evaluated at inter-sections of fixed and moving geometric objects.

2 Elementary mean value formulae

We begin with the second part of the program, the development of meanvalue formulae for fixed and moving geometric objects. By “moving” wemean here that the geometric objects, which are in Euclidean space, un-dergo translations or rigid motions. The mean values will be taken with

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respect to invariant measures on the groups of translations or rigid mo-tions. The present section is still part of the introduction and will discussa few elementary examples of such mean value formulae.

We work in n-dimensional Euclidean space Rn (n ≥ 2). The subsetsof Rn which will later (in dimensions two and three) be used to modelreal material, should not be too complicated, in order that functionals likesurface area or Euler characteristic are defined (locally). It is sufficient forpractical applications to consider only sets which can locally be representedas finite unions of convex bodies (non-empty, compact convex sets). Webegin by considering only convex bodies; it will later be easy to extend theresults to more general sets of the type just described. By Kn we denotethe set of convex bodies in Rn.

The following is a basic example of the type of questions that we willhave to answer. Let K,M ∈ Kn be two convex bodies. Let M undergotranslations, that is, we consider M + t for t ∈ Rn. What is the mean valueof the volume of K ∩ (M + t), taken over all t with K ∩ (M + t) 6= ∅? Themean value here refers to the invariant measure on the translation group,which can be identified with the Lebesgue measure λ on Rn. For convexbodies K, we write Vn(K) = λ(K) for the volume. Thus we are asking forthe mean value ∫

Rn Vn(K ∩ (M + t)) dλ(t)∫Rn χ(K ∩ (M + t)) dλ(t)

. (1)

Note that χ(K ′) = 1 for a non-empty convex body K ′ and χ(∅) = 0, so thatthe denominator is indeed the total measure of all translation vectors t forwhich K ∩ (M + t) 6= ∅. Thus we have to determine integrals of the type∫

Rn

ϕ(K ∩ (M + t)) dλ(t)

for different functionals ϕ. Extensions of this problem will be our mainconcern in these lectures.

It is not difficult to determine the numerator in (1). Denoting the indi-cator function of a set A ⊂ Rn by 1A, we have

Vn(K ∩ (M + t)) =∫

Rn

1K∩(M+t)(x) dλ(x)

and1K∩(M+t)(x) = 1K(x)1M+t(x)

with

1M+t(x) = 1 ⇔ x ∈M + t ⇔ t ∈M∗ + x ⇔ 1M∗+x(t) = 1.

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Here we have denoted by

M∗ := y ∈ Rn : −y ∈M

the set obtained from M by reflection in the origin. Now Fubini’s theoremgives ∫

Rn

Vn(K ∩ (M + t)) dλ(t)

=∫

Rn

∫Rn

1K∩(M+t)(x) dλ(x) dλ(t)

=∫

Rn

∫Rn

1K(x)1M∗+x(t) dλ(t) dλ(x)

=∫

Rn

1K(x)Vn(M∗ + x) dλ(x)

= Vn(M∗)∫

Rn

1K(x) dλ(x)

and hence ∫Rn

Vn(K ∩ (M + t)) dλ(t) = Vn(K)Vn(M). (2)

Note that we have used the invariance of the volume under translationsand reflections.

The denominator in (1) is of a different type. We have

χ(K ∩ (M + t)) = 1 ⇔ K ∩ (M + t) 6= ∅⇔ ∃ k ∈ K ∃m ∈M : k = m+ t

⇔ t = k −m with k ∈ K,m ∈M⇔ t ∈ K +M∗

⇔ 1K+M∗(t) = 1

and hence ∫Rn

χ(K ∩ (M + t)) dλ(t) = Vn(K +M∗). (3)

Convex geometry tells us that

Vn(K +M∗) =n∑

i=0

(n

i

)V (K, . . . ,K︸ ︷︷ ︸

i

,M∗, . . . ,M∗︸ ︷︷ ︸n−i

),

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where the function V : (Kn)n → R is the so-called mixed volume. Theessential observation for us is here that the obtained expression cannot besimplified further. In particular, there is no separation of the roles of K andM on the right-hand side, as it occurred in (2). Such a separation is onlyachieved if we integrate, not only over the translations of M as in (3), butover all rigid motions of M . This will be one of the fundamental results ofintegral geometry to be obtained later.

For the moment, however, we stay with the translation group alone.The idea leading to (2) can be extended, to give a first general formula oftranslative integral geometry.

When we talk of a measure on a locally compact space E, we alwaysmean a non-negative, countably additive, extended real-valued function onthe σ-algebra B(E) of Borel sets of E. Such a measure is called locally finiteif it is finite on compact sets.

2.1 Theorem. Let α be a locally finite measure on Rn, and let A,B ∈B(Rn). Then ∫

Rn

α(A ∩ (B + t)) dλ(t) = α(A)λ(B). (4)

Proof. Using Fubini’s theorem, we obtain∫Rn

α(A ∩ (B + t)) dλ(t)

=∫

Rn

∫Rn

1A∩(B+t)(x) dα(x) dλ(t)

=∫

Rn

∫Rn

1A(x)1B+t(x) dλ(t) dα(x)

=∫

Rn

1A(x)∫

Rn

1B∗+x(t) dλ(t) dα(x)

=∫

Rn

1A(x)λ(B∗ + x) dα(x)

= α(A)λ(B).

This can be used to obtain a counterpart to the translative integral formula(2), with volume replaced by surface area. First we have to explain what we

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mean by the surface area of a general convex body, which need not satisfyany smoothness assumptions. For that purpose, let us first recall the notionof the p-dimensional Hausdorff measure, for p ≥ 0.

We equip Rn with the usual scalar product 〈·, ·〉 and the induced norm‖ · ‖. For a subset G ⊂ Rn, the diameter is defined by

D(G) := sup‖x− y‖ : x, y ∈ G.

Now for an arbitrary subset M and for δ > 0 one defines

Hpδ(M) :=

πp/2

2pΓ(1 + p2 )

inf

∞∑i=1

D(Gi)p : (Gi)i∈N sequence of open sets

with D(Gi) ≤ δ and M ⊂∞⋃

i=1

Gi

.

The limitHp(M) := lim

δ→0+Hp

δ(M) = supδ>0

Hpδ(M)

exists in R∪∞ and is called the p-dimensional (outer) Hausdorff measureof M . The restriction of Hp to the σ-algebra B(Rn) of Borel sets is ameasure. One can show that Hn(A) = λ(A) for A ∈ B(Rn).

Now the surface area of a convex body K ∈ Kn with interior points isdefined by

Hn−1(∂K) =: 2Vn−1(K),

where ∂ denotes the boundary. The notation 2Vn−1 is chosen with respectto later developments. For K ∈ Kn without interior points, we defineVn−1(K) := Hn−1(K). This is zero if K is of dimension less than n− 1.

2.2 Theorem. Let K,M ∈ Kn be convex bodies with interior points. Then∫Rn

Vn−1(K ∩ (M + t)) dλ(t) = Vn−1(K)Vn(M) + Vn(K)Vn−1(M). (5)

Proof. The boundary of the intersection K ∩ (M + t) consists of two parts:

∂(K ∩ (M + t)) = [∂K ∩ (M + t)] ∪ [K ∩ (∂M + t)].

The intersection of the two sets on the right satisfies

[∂K ∩ (M + t)] ∩ [K ∩ (∂M + t)] ⊂ ∂K ∩ (∂M + t).

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We defineα(A) := Hn−1(∂K ∩A) for A ∈ B(Rn).

Then α is a finite measure on Rn. From (4) (with A = ∂K and B = ∂M)we get ∫

Rn

Hn−1(∂K ∩ (∂M + t)) dλ(t) = Hn−1(∂K)λ(∂M) = 0.

Since the integrand is nonnegative, it follows that

Hn−1(∂K ∩ (∂M + t)) = 0 for λ-almost all t,

that is, for all t ∈ Rn \N , with some set N satisfying λ(N) = 0. Hence, forall t ∈ Rn \N we have

Hn−1(∂(K ∩ (M + t)) = Hn−1(∂K ∩ (M + t)) +Hn−1(K ∩ (∂M + t)). (6)

Using (4) with A = ∂K and B = M , we further obtain∫Rn

Hn−1(∂K ∩ (M + t)) dλ(t) = Hn−1(∂K)λ(M).

Moreover, ∫Rn

Hn−1(K ∩ (∂M + t)) dλ(t)

=∫

Rn

Hn−1((K − t) ∩ ∂M) dλ(t)

=∫

Rn

Hn−1(∂M ∩ (K + t)) dλ(t)

= Hn−1(∂M)λ(K).

Here we have used the facts that Hn−1 is translation invariant and that theLebesgue measure is invariant under the inversion t 7→ −t. Finally we haveused (4) again.

Since equation (6) holds for all t ∈ Rn \N and since the null set N canbe neglected in the integration, we deduce that∫

Rn

Hn−1(∂(K ∩ (M + t)) dλ(t) = Hn−1(∂K)λ(M) +Hn−1(∂M)λ(K).

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This is precisely the assertion (5).

Instead of intersecting a fixed convex body with a translated one, we nowbriefly consider the intersections with a translated hyperplane. We param-eterize hyperplanes in the form

H(u, τ) := x ∈ Rn : 〈x, u〉 = τ

with a unit vector u ∈ Rn and a real number τ ∈ R. Thus u is one of thetwo unit normal vectors of the (unoriented) hyperplane H(u, τ).

For a convex body K ∈ Kn, Fubini’s theorem immediately gives∫R

Vn−1(K ∩H(u, τ)) dτ = Vn(K).

Can we obtain the surface area of a convex body K ∈ Kn with interiorpoints in a similar way, that is, by a formula of type∫

Rn

Hn−2(∂K ∩H(u, τ)) dτ = cnVn−1(K)

with some constant cn? Simple examples (balls and cubes in R3) show thatsuch a formula does not hold with a constant independent of K. However,we shall later see that∫

Sn−1

∫Rn

Hn−2(∂K ∩H(u, τ)) dτ dσ(u) = cnVn−1(K) (7)

does hold with a constant cn. Here the outer integration is over the unitsphere Sn−1 with respect to the rotation invariant measure σ.

Both integrations in (7) together can be interpreted as one integrationover the space of hyperplanes, with respect to a rigid motion invariant mea-sure on that space. Thus we have now two examples for the simplifyingeffect in obtaining mean values when the integrations are performed withrespect to motion invariant measures. This observation will be considerablyelaborated in the following.

3 Invariant measures of Euclidean geometry

Integral geometry is based on the notion of invariant measure. Here invari-ance refers to a group operation and thus to a homogeneous space. Invariant

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measures on homogeneous spaces are also known as Haar measures. We donot presuppose here any knowledge of the theory of Haar measure. In thepresent section, we give an elementary introduction to the invariant mea-sures on the groups and homogeneous spaces that are used in the integralgeometry of Euclidean space.

A topological group is a group G together with a topology on G suchthat the map from G × G to G defined by (x, y) 7→ xy and the map fromG to G defined by x 7→ x−1 are continuous. Let G be a group and X anon-empty set. An operation of G on X is a map ϕ : G×X → X satisfying

ϕ(g, ϕ(g′, x)) = ϕ(gg′, x), ϕ(e, x) = x

for all g, g′ ∈ G, the unit element e of G and all x ∈ X. One also saysthat G operates on X, by means of ϕ. For ϕ(g, x) one usually writes gx,provided that the operation is clear from the context. The group G operatestransitively on X if for any x, y ∈ X there exists g ∈ G so that y = gx. IfG is a topological group, X is a topological space, and the operation ϕ iscontinuous, one says that G operates continuously on X.

The following situation often occurs: X is a nonempty set and G is agroup of transformations (bijective mappings onto itself) of X, with thecomposition as group multiplication; the operation of G on X is given by(g, x) 7→ gx := image of x under g. When transformation groups occur inthe following, multiplication and operation are always understood in thissense.

We consider three groups of bijective affine maps of Rn onto itself, thetranslation group Tn, the rotation group SOn, and the rigid motion groupGn. The translations t ∈ Tn are the maps of the form t = tx with x ∈ Rn,where tx(y) := y+x for y ∈ Rn. The mapping τ : x 7→ tx is an isomorphismof the additive group Rn onto Tn. Hence, we can identify Tn with Rn, whichwe shall often do tacitly. In particular, Tn carries the topology inheritedfrom Rn via τ . Since txty = tx+y and t−1

x = t−x, composition and inversionare continuous, hence Tn is a topological group. In view of the topologicalproperties of Rn we can thus state the following.

3.1 Theorem. The translation group Tn is an abelian, locally compacttopological group with countable base. The operation of Tn on Rn is contin-uous.

The elements of the rotation group SOn are the linear mappings ϑ : Rn →Rn that preserve scalar product and orientation; they are called (proper)rotations. With respect to the standard (orthonormal) basis of Rn, everyrotation ϑ is represented by an orthogonal matrix M(ϑ) with determinant 1.

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The mapping µ : ϑ 7→ M(ϑ) is an isomorphism of the group SOn onto thegroup SO(n) of orthogonal (n, n)-matrices with determinant 1 under matrixmultiplication. If we identify an (n, n)-matrix with the n2-tuple of its entries(in lexicographic order, say), we can consider SO(n) as a subset of Rn2

. Thisset is bounded, since the rows of an orthogonal matrix are normalized, andit is closed in Rn2

, hence compact. The mappings (M,N) 7→ MN andM 7→ M−1 are continuous, and so is the mapping (M,x) 7→ Mx (wherex is considered as an (n, 1)-matrix) from SO(n) × Rn into Rn. Using themapping µ−1 to transfer the topology from SO(n) to SOn, we thus obtainthe following.

3.2 Theorem. The rotation group SOn is a compact topological group withcountable base. The operation of SOn on Rn is continuous.

The elements of the motion group Gn are the affine maps g : Rn → Rn

that preserve distances between points and the orientation; they are called(rigid) motions. Every rigid motion g ∈ Gn can be represented uniquely asthe composition of a rotation ϑ and a translation tx, that is, g = tx ϑ, orgy = ϑy + x for y ∈ Rn. The mapping

γ : Rn × SOn → Gn

(x, ϑ) 7→ tx ϑ

is bijective. We use it to transfer the topology from Rn×SOn to Gn. UsingTheorems 3.1 and 3.2, it is then easy to show the following.

3.3 Theorem. Gn is a locally compact topological group with countablebase. Its operation on Rn is continuous.

After these topological groups, we now consider the homogeneous spacesthat will play a role in the following. Let q ∈ 0, . . . , n, let Ln

q be theset of all q-dimensional linear subspaces of Rn, and let En

q be the set ofall q-dimensional affine subspaces of Rn. The natural operation of SOn onLn

q is given by (ϑ,L) 7→ ϑL := image of L under ϑ. Similarly, the naturaloperation of Gn on En

q is given by (g,E) 7→ gE := image of E under g. Weintroduce suitable topologies on Ln

q and Enq . For this, let Lq ∈ Ln

q be fixedand let L⊥q be its orthogonal complement. The mappings

βq : SOn → Lnq

ϑ ϑLq

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andγq : L⊥q × SOn → En

q

(x, ϑ) 7→ ϑ(Lq + x)

are surjective (but not injective). We endow Lnq with the finest topology

for which βq is continuous, and Enq with the finest topology for which γq

is continuous. Thus a subset A ∈ Enq , for example, is open if and only if

γ−1q (A) is open. It is an elementary task to prove the following.

3.4 Theorem. Lnq is compact and has a countable base, the map βq is

open, and the operation of SOn on Lnq is continuous and transitive.

3.5 Theorem. Enq is locally compact and has a countable base, the map γq

is open, and the operation of Gn on Enq is continuous and transitive.

It should be remarked that the topologies on Lnq and En

q , as well as theinvariant measures on these spaces to be introduced below, do not dependon the special choice of the subspace Lq. This follows easily from the factthat SOn operates transitively on Ln

q , and Gn operates transitively on Enq .

The topological spaces Lnq are called Grassmann manifolds; a common

notation for Lnq is G(n, q). The spaces En

q are also called affine Grassman-nians.

Occasionally, we have talked of homogeneous spaces; it seems, therefore,appropriate here to give the general definition. If G is a topological group,a homogeneous G-space is, by definition, a pair (X,ϕ), where X is a topo-logical space and ϕ is a transitive continuous operation of G on X with theadditional property that the map ϕ(·, p) is open for p ∈ X. In this sense,Ln

q is a homogeneous SOn-space (with respect to the standard operation),and En

q is a homogeneous Gn-space. Also with the standard operations, Rn

is a homogeneous Tn-space and Gn-space, and the unit sphere

Sn−1 := x ∈ Rn : ‖x‖ = 1

is a homogeneous SOn-space.We shall now introduce invariant measures on the groups and homo-

geneous spaces considered. We begin with some general definitions andremarks. All topological spaces occurring here are locally compact and sec-ond countable. By a Borel measure ρ on X we understand a measure on theσ-algebra B(X) of Borel sets of X satisfying ρ(K) < ∞ for every compactset K ⊂ X. Every such measure is regular. Instead of ‘Borel measure’we often say ‘measure’ for short. The notion ‘measurable’, without extraspecification, means ‘Borel measurable’.

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Let the topological group G operate continuously on the space X. Ameasure ρ on X is called G-invariant (or briefly invariant, if G is clear fromthe context) if

ρ(gA) = ρ(A) for all A ∈ B(X) and all g ∈ G.

This definition makes sense: for each g ∈ G, the mapping x 7→ gx is ahomeomorphism, hence A ∈ B(X) implies gA ∈ B(X). Invariant regularBorel measures on locally compact homogeneous spaces are called Haarmeasures, if they are not identically zero.

From basic measure theory, we assume familiarity with Lebesgue mea-sure on Rn, in particular with the following result. Here we use the unitcube Cn := [0, 1]n for normalization.

3.6 Theorem and Definition. There is a unique translation invariantmeasure λ on B(Rn) satisfying λ(Cn) = 1. It is called the Lebesgue measure.

It is easy to see that λ is also rotation invariant (SOn-invariant). If ϑ ∈ SOn

and if one defines ρ(A) := λ(ϑA) for A ∈ B(Rn), then ρ is a translationinvariant measure on B(Rn). By Theorem 3.6, ρ = cλ with c = ρ(Cn). Theunit ball Bn satisfies cλ(Bn) = ρ(Bn) = λ(ϑBn) = λ(Bn), hence c = 1.

Since the Lebesgue measure λ is thus rigid motion invariant, it is theHaar measure on the homogeneous Gn-space Rn, normalized in a specialway.

We mention the special value

κn := λ(Bn) =π

n2

Γ(1 + n2 ),

which will play a role in many later formulae. We put κ0 := 1.The Haar measure on the homogeneous SOn-space Sn−1, the unit sphere,

is easily derived from the Lebesgue measure. For A ∈ B(Sn−1) we define

A := αx ∈ Rn : x ∈ A, 0 ≤ α ≤ 1.

A standard argument shows that A ∈ B(Rn), hence we can define σ(A) :=nλ(A). This yields a finite measure σ on B(Sn−1) for which

σ(Sn−1) =: ωn = nκn =2π

n2

Γ(n2 ).

The rotation invariance of λ implies the rotation invariance of σ. We callσ, with the normalization specified above, the spherical Lebesgue measure.

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Up to a constant factor, σ is the only rotation invariant Borel measure onB(Sn−1). This follows from Corollary 3.12 below.

Our next aim is the introduction of an invariant measure on the rotationgroup SOn. For a measure on a group, several notions of invariance arenatural. A topological group G operates on itself by means of the mapping(g, x) 7→ gx (multiplication in G). The corresponding invariance on G iscalled left invariance. More generally, for g ∈ G and A ⊂ G we write

gA := ga : a ∈ A, Ag := ag : a ∈ A, A−1 := a−1 : a ∈ A.

If A ∈ B(G), then also gA, Ag, A−1 are Borel sets. A measure ρ on G iscalled left invariant if ρ(gA) = ρ(A), and right invariant if ρ(Ag) = ρ(A),for all A ∈ B(G) and all g ∈ G. The measure ρ is inversion invariant ifρ(A−1) = ρ(A) for all A ∈ B(G). If ρ has all three invariance properties, itis just called invariant.

With these definitions we connect two general remarks. Let ρ be aleft invariant measure on the topological group G. Then each measurablefunction f ≥ 0 on G satisfies∫

G

f(ag) dρ(g) =∫G

f(g) dρ(g) (8)

for all a ∈ G. This follows immediately from the definition of the integral.Vice versa, if (8) holds for all measurable functions f ≥ 0, then the leftinvariance of ρ is obtained by applying (8) to indicator functions. Similarly,the right invariance of ρ is equivalent to∫

G

f(ga) dρ(g) =∫G

f(g) dρ(g) (9)

for a ∈ G, and the inversion invariance of ρ is equivalent to∫G

f(g−1) dρ(g) =∫G

f(g) dρ(g), (10)

in each case for all measurable functions f ≥ 0.The following theorem on invariant measures on compact groups will be

needed for the rotation group only, but can be proved without additionaleffort in a more general setting.

3.7 Theorem. Every left invariant Borel measure on a compact group withcountable base is invariant.

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Proof. Let ν be a left invariant Borel measure on the group G satisfying theassumptions. Since it is finite on compact sets, we may assume ν(G) = 1,without loss of generality. For measurable functions f ≥ 0 on G and forx ∈ G we have∫

f(y−1x) dν(y) =∫f((x−1y)−1) dν(y) =

∫f(y−1) dν(y). (11)

Here the integrations extend over all of G; similar conventions will beadopted in the following. Fubini’s theorem gives∫

f(y−1) dν(y) =∫ ∫

f(y−1x) dν(y) dν(x)

=∫ ∫

f(y−1x) dν(x) dν(y) =∫f(x) dν(x).

Hence, the measure ν is inversion invariant. Using this fact and (11), weget for x ∈ G that ∫

f(yx) dν(y) =∫f(y−1x) dν(y)

=∫f(y−1) dν(y) =

∫f(y) dν(y),

which shows that ν is also right invariant.

Concerning the application of Fubini’s theorem here and later, we remarkthe following. All topological spaces occurring in our considerations are lo-cally compact and second countable, thus they are σ-compact. Moreover,all the measures that occur are finite on compact sets. Therefore, all mea-sure spaces under consideration are σ-finite, so that Fubini’s theorem canbe applied in its usual form.

The following uniqueness result for invariant measures makes specialassumptions, but in this form it is sufficient for our purposes and is easy toprove.

3.8 Theorem. Let G be a locally compact group with a countable base, letν 6= 0 be an invariant and µ a left invariant Borel measure on G. Thenµ = cν with a constant c ≥ 0.

Proof. For measurable functions f, g ≥ 0 on G we have∫f dν

∫g dµ =

∫ ∫f(xy)g(y) dν(x) dµ(y)

16

=∫ ∫

f(xy)g(y) dµ(y) dν(x) =∫ ∫

f(y)g(x−1y) dµ(y) dν(x)

=∫f(y)

∫g(x−1y) dν(x) dµ(y) =

∫g dν

∫f dµ.

Here we have used, besides Fubini’s theorem, the right and inversion invari-ance of ν and the left invariance of µ.

Since ν 6= 0, there is a compact set A0 ⊂ G with ν(A0) > 0. Forarbitrary A ∈ B(G) we put f := 1A0 and g := 1A and obtain ν(A0)µ(A) =ν(A)µ(A0), hence µ = cν with c := µ(A0)/ν(A0).

The notation 1A used here for the indicator function of a set A will also beemployed in the following.

Now we turn to the existence of some invariant measures. First we de-scribe a direct construction of the invariant measure on the rotation group,without recourse to the general theory of Haar measure.

3.9 Theorem. On the rotation group SOn, there is an invariant measureν with ν(SOn) = 1.

Proof. By LIn we denote the set of linearly independent n-tuples of vec-tors from the unit sphere Sn−1. We define a map ψ : LIn → SOn inthe following way. Let (x1, . . . , xn) ∈ LIn. By Gram-Schmidt orthonor-malization, we transform (x1, . . . , xn) into the n-tuple (z1, . . . , zn); then wedenote by (z1, . . . , zn) the positively oriented n-tuple for which zi := zi fori = 1, . . . , n−1 and zn := ±zn. If (e1, . . . , en) denotes the standard basis ofRn, there is a unique rotation ϑ ∈ SOn satisfying ϑei = zi for i = 1, . . . , n.We define ψ(x1, . . . , xn) := ϑ.

Explicitly, we have zi = yi/‖yi‖ with y1 = x1 and

yk = xk −k−1∑j=1

〈xk, yj〉yj

‖yj‖2, k = 2, . . . , n.

From this representation, the following is evident. If ρ ∈ SOn is a rota-tion and if the n-tuple (x1, . . . , xn) ∈ LIn is transformed into (z1, . . . , zn)and then into (z1, . . . , zn), then the n-tuple (ρx1, . . . , ρxn) is transformedinto (ρz1, . . . , ρzn) and subsequently into (ρz1, . . . , ρzn). Thus we haveψ(ρx1, . . . , ρxn) = ρψ(x1, . . . , xn).

For (x1, . . . , xn) ∈ (Sn−1)n \ LIn we define ψ(x1, . . . , xn) := id. For theproduct measure

σ⊗n := σ ⊗ · · · ⊗ σ︸ ︷︷ ︸n

,

17

the set (Sn−1)n \ LIn has measure zero; hence for any ρ ∈ SOn the equal-ity ψ(ρx1, . . . , ρxn) = ρψ(x1, . . . , xn) holds σ⊗n-almost everywhere. Themapping ψ : (Sn−1)n → SOn is measurable, since LIn is open and ψ iscontinuous on LIn and constant on (Sn−1)n \ LIn.

Now we define ν as the image measure of σ⊗n under ψ, thus ν = ψ(σ⊗n).Then ν is a finite measure on SOn, and for ρ ∈ SOn and measurable f ≥ 0we obtain ∫

SOn

f(ρϑ) dν(ϑ)

=∫

(Sn−1)n

f(ρψ(x1, . . . , xn)) dσ⊗n(x1, . . . , xn)

=∫

(Sn−1)n

f(ψ(ρx1, . . . , ρxn)) dσ⊗n(x1, . . . , xn)

=∫

Sn−1

· · ·∫

Sn−1

f(ψ(ρx1, . . . , ρxn)) dσ(x1) · · · dσ(xn)

=∫

Sn−1

· · ·∫

Sn−1

f(ψ(x1, . . . , xn)) dσ(x1) · · · dσ(xn)

=∫

SOn

f(ϑ) dν(ϑ).

Here we have used the rotation invariance of the spherical Lebesgue measure.We have proved that the measure ν is left invariant and thus invariant, byTheorem 3.7. The measure ν := ν/ν(SOn) is invariant and normalized.

From now on, ν will always denote the normalized invariant measure onSOn.Now we turn to the motion group Gn. Since it is not compact, an invariantmeasure µ on Gn cannot be finite. In order to normalize µ, we specify thecompact set A0 := γ(Cn × SOn) and require that µ(A0) = 1.

3.10 Theorem. On the motion group Gn, there is an invariant measureµ with µ(A0) = 1. Up to a constant factor, it is the only left invariantmeasure on Gn.

Proof. We define µ as the image measure of the product measure λ ⊗ νunder the homeomorphism γ : Rn × SOn → Gn defined by (8). Then µ isa Borel measure on Gn with µ(γ(Cn × SOn)) = λ(Cn)ν(SOn) = 1.

18

To show the left invariance of µ, let f ≥ 0 be a measurable function onGn and let g′ ∈ Gn. With g′ = γ(t′, ϑ′) we have∫

Gn

f(g′g) dµ(g)

=∫

SOn

∫Rn

f(γ(t′, ϑ′)γ(t, ϑ)) dλ(t) dν(ϑ)

=∫

SOn

∫Rn

f(γ(t′ + ϑ′t, ϑ′ϑ)) dλ(t) dν(ϑ)

=∫

SOn

∫Rn

f(γ(t, ϑ)) dλ(t) dν(ϑ)

=∫

Gn

f(g) dµ(g),

where we have used the motion invariance of λ and the left invariance of ν.Hence, µ is left invariant. Similarly, the right invariance of ν implies via∫

Gn

f(gg′) dµ(g) =∫

SOn

∫Rn

f(γ(t+ ϑt′, ϑϑ′)) dλ(t) dν(ϑ)

=∫

SOn

∫Rn

f(γ(t, ϑ)) dλ(t) dν(ϑ) =∫

Gn

f(g) dµ(g)

the right invariance of µ. The inversion invariance of µ is obtained from∫Gn

f(g−1) dµ(g) =∫

SOn

∫Rn

f(γ(−ϑ−1t, ϑ−1)) dλ(t) dν(ϑ)

=∫

SOn

∫Rn

f(γ(t, ϑ)) dλ(t) dν(ϑ) =∫

Gn

f(g) dµ(g),

where the inversion invariance of ν was used.The uniqueness assertion is a special case of Theorem 3.8.

Having constructed invariant measures on the groups SOn and Gn, we nextturn to the introduction of invariant measures on the homogeneous spacesLn

q and Enq . First we prove a formula of integral-geometric type, extending

Theorem 2.1, which will be useful for obtaining uniqueness results.

19

3.11 Theorem. Suppose that the compact group G operates continuouslyand transitively on the Hausdorff space X, and that G and X have countablebases. Let ν be an invariant measure on G with ν(G) = 1, let ρ 6= 0 be aG-invariant Borel measure on X and α an arbitrary Borel measure on X.Then ∫

G

α(A ∩ gB) dν(g) = α(A)ρ(B)/ρ(X)

for all A,B ∈ B(X).

Proof. If ϕ denotes the operation of G on X and if x ∈ X, the mappingϕ(·, x) : G → X is continuous and surjective, hence X is compact. There-fore, the Borel measures α and ρ are finite. Let A,B ∈ B(X) and g ∈ G begiven, then

α(A ∩ gB) =∫X

1A∩gB dα(x) =∫X

1A(x)1B(g−1x) dα(x).

Fubini’s theorem yields∫G

α(A ∩ gB) dν(g) =∫X

1A(x)∫G

1B(g−1x) dν(g) dα(x). (12)

The integral∫G

1B(g−1x) dν(g) does not depend on x, since for y ∈ X there

exists g ∈ G with y = gx and, therefore,∫G

1B(g−1y) dν(g) =∫G

1B((g−1g)−1x) dν(g) =∫G

1B(g−1x) dν(g).

Hence we obtain

ρ(X)∫G

1B(g−1x) dν(g) =∫X

1B(g−1x) dν(g) dρ(x)

=∫G

∫X

1B(g−1x) dρ(x) dν(g) =∫G

ρ(gB) dν(g) = ρ(B).

Inserting this into (12), we complete the proof.

3.12 Corollary. Suppose that the compact group G operates continuouslyand transitively on the Hausdorff space X and that G and X have countablebases. Let ν be an invariant measure on G with ν(G) = 1.

20

Then there exists a unique G-invariant measure ρ on X with ρ(X) = 1.It can be defined by

ρ(B) := ν(g ∈ G : gx0 ∈ B), B ∈ B(X),

with arbitrary x0 ∈ X.

Proof. Let ρ be a G-invariant measure on X with ρ(X) = 1. We choosex0 ∈ X and let α be the Dirac measure on X concentrated in x0. Theorem3.11 with A := x0 gives

ρ(B) = ν(g ∈ G : g−1x0 ∈ B)

for B ∈ B(X). Thus ρ is unique. Vice versa, if ρ is defined in this way, it isclear that it is a G-invariant normalized measure.

Now we turn to invariant measures on the space Lnq of q-dimensional linear

subspaces and on the space Enq of q-dimensional affine subspaces. By an

invariant measure on Lnq we understand an SOn-invariant measure on Ln

q ,and an invariant measure on En

q is defined as a Gn-invariant measure on Enq .

3.13 Theorem. On Lnq there is a unique invariant measure νq, normalized

by νq(Lnq ) = 1.

This is just a special case of Corollary 3.12. We also notice that νq is theimage measure of ν under the map βq defined by (8).

3.14 Theorem. On Enq there is an invariant measure µq. It is unique up

to a constant factor.

Proof. We recall that we have chosen a subspace Lq ∈ Lnq and defined the

map γq : L⊥q × SOn → Enq by (8). Let λ(n−q) be Lebesgue measure on L⊥q .

We defineµq := γq(λ(n−q) ⊗ ν), (13)

so that µq is the image measure of the product measure λ(n−q) ⊗ ν underthe map γq. If A ⊂ En

q is compact, the sets

γq(x ∈ L⊥q : ‖x‖ < k × SOn), k ∈ N,

constitute an open covering of A, hence A is included in one of these sets.It follows that µq(A) <∞.

21

By the definition of µq, integrals with respect to µq can be expressed inthe following way. For a nonnegative measurable function f on En

q ,∫En

q

f dµq =∫

SOn

∫L⊥q

f(ρ(Lq + x)) dλ(n−q)(x) dν(ρ)

=∫

SOn

∫(ρLq)⊥

f(ρLq + y) dλ(n−q)(y) dν(ρ).

Since the invariant measure νq on Lnq is the image measure under the map

βq, this can be written as∫En

q

f dµq =∫Ln

q

∫L⊥

f(L+ y) dλ(n−q)(y) dνq(L). (14)

From this representation we infer that µq does not depend on the choice ofthe subspace Lq.

To show the invariance of µq, let g = γ(x, ϑ) ∈ G and let f ≥ 0 be ameasurable function on En

q . Denoting by Π the orthogonal projection ontoL⊥q , we have∫

Enq

f(gE) dµq(E)

=∫

SOn

∫L⊥q

f(gρ(Lq + y)) dλ(n−q)(y) dν(ρ)

=∫

SOn

∫L⊥q

f(ϑρ(Lq + y + Π(ρ−1ϑ−1x))) dλ(n−q)(y) dν(ρ)

=∫

SOn

∫L⊥q

f(ϑρ(Lq + y)) dλ(n−q)(y) dν(ρ)

=∫

SOn

∫L⊥q

f(ρ(Lq + y)) dλ(n−q)(y) dν(ρ)

=∫En

q

f(E) dµq(E),

where we have used the invariance properties of λ(n−q) and ν. This showsthe invariance of µq.

22

To prove the uniqueness (up to a factor), we assume that τ is anotherinvariant Borel measure on En

q . Let Lnq (respectively En

q ) be the open set ofall L ∈ Ln

q (respectively E ∈ Enq ) that intersect L⊥q in precisely one point.

The mappingδq : L⊥q × Ln

q → Enq

(x, L) 7→ L+ x

is a homeomorphism. For fixed B ∈ B(Lnq ) and arbitrary A ∈ B(L⊥q ) we

define η(A) := τ(δq(A × B)). Then η is a Borel measure on L⊥q , which isinvariant under the translations of L⊥q into itself. Theorem 3.6 implies thatη(A) = λ(n−q)(A)α(B) with a constant α(B) ≥ 0. Hence we have

τ(δq(A×B)) = λ(n−q)(A)α(B)

for arbitrary A ∈ B(L⊥q ) and B ∈ B(Lnq ). Obviously this equation defines

a finite measure α on B(Lnq ), and δ−1

q (τ) = λ(n−q) ⊗ α. For a measurablefunction f ≥ 0 on En

q we obtain∫En

q

f dτ =∫Ln

q

∫L⊥q

f(L+ x) dλ(n−q)(x) dα(L)

=∫Ln

q

∫L⊥

f(L+ y) dλ(n−q)(y) dϕ(L) (15)

with a measure ϕ on Lnq defined by dϕ(L)/dα(L) = D(L⊥q , L

⊥)−1, whereD(L⊥q , L

⊥) is the absolute determinant of the orthogonal projection fromL⊥q onto L⊥.

Now let B ∈ B(Lnq ) and

B′ := L+ y : L ∈ B, y ∈ L⊥ ∩Bn.

By β(B) := τ(B′) we define a rotation invariant finite measure β on Lnq .

According to Theorem 3.13 it is a multiple of νq. On the other hand, (15)gives τ(B′) = κn−qϕ(B) for B ⊂ Ln

q . Hence, there is a constant c withϕ(B) = cνq(B) for all Borel sets B ⊂ Ln

q . From (15) and (14) we deducethat τ(A) = cµq(A) for all Borel sets A ⊂ En

q . Since µq does not depend onthe choice of the subspace Lq ∈ Ln

q , it is easy to see that τ = cµq.

By its definition, the measure µq comes with a particular normalization.We want to determine the measure of all q-flats meeting the unit ball Bn.

23

SinceE ∈ En

q : E ∩Bn 6= ∅ = γq((Bn ∩ L⊥q )× SOn),

we getµq(E ∈ En

q : E ∩Bn 6= ∅) = κn−q.

For r > 0 we have

µq(E ∈ Enq : E ∩ rBn 6= ∅) = rn−qκn−q.

4 Additive functionals

Beside special Haar measures, another type of invariant measures that wewill use are finitely additive measures on certain systems of subsets of Eu-clidean space.

We begin with some general definitions. Let ϕ be a function on a familyS of sets with values in some abelian group. The function ϕ is called additiveor a valuation if

ϕ(K ∪ L) + ϕ(K ∩ L) = ϕ(K) + ϕ(L) (16)

holds whenever K,L ∈ S are sets such that also K ∪L ∈ S and K ∩L ∈ S.If ∅ ∈ S, one also assumes that ϕ(∅) = 0. We say that the system S is ∩-stable (intersection stable) if K,L ∈ S implies K∩L ∈ S∪∅. In this case,we denote by U(S) the system of all finite unions of sets in S (including theempty set). The system U(S) is closed under finite unions and intersectionsand thus is a lattice.

Now let ϕ be an additive function on S. One may ask whether it has anextension to an additive function on the lattice U(S). Suppose that suchan extension exists, and denote it also by ϕ. Then for K1, . . . ,Km ∈ U(S)the formula

ϕ(K1 ∪ · · · ∪Km) =m∑

r=1

(−1)r−1∑

i1<···<ir

ϕ(Ki1 ∩ · · · ∩Kir ) (17)

holds. For m = 2, this is just the equation (16) defining additivity. Thegeneral case of (17) is easily obtained by induction. This formula is calledthe inclusion-exclusion principle.

Formula (17) shows that an additive extension from the ∩-stable sys-tem S to the generated lattice U(S), if it exists, is uniquely determined.Conversely, however, one cannot just use (17) for the definition of suchan extension, since the representation of an element of U(S) in the form

24

K1 ∪ · · · ∪Km with Ki ∈ S is in general not unique. Hence, the existenceof an additive extension, if there is one, must be proved in a different way.

We will write (17) in a more concise form. For m ∈ N, let S(m) denotethe set of all non-empty subsets of 1, . . . ,m. For v ∈ S(m), let |v| :=card v. If K1, . . . ,Km are given, we write

Kv := Ki1 ∩ · · · ∩Kimfor v = i1, . . . , ir ∈ S(m).

With these conventions, the inclusion-exclusion principle (17) can be writtenin the form

ϕ(K1 ∪ · · · ∪Km) =∑

v∈S(m)

(−1)|v|−1ϕ(Kv). (18)

Of considerable importance in the following is the lattice U(Kn) gener-ated by the ∩-stable family Kn ∪ ∅. Thus the system U(Kn) consists ofall subsets of Rn that can be represented as finite unions of convex bodies.We call such sets polyconvex (following Klain-Rota [4], who in turn followedE. de Giorgi). Hadwiger [2] used for U(Kn) the name ‘Konvexring’, whichhas been translated (perhaps not so luckily) into convex ring.

The simplest non-zero valuation on Kn is given by χ(K) = 1 for allK ∈ Kn. We show that it has an additive extension to U(Kn).

4.1 Theorem. There is a unique valuation χ on the convex ring U(Kn)satisfying

χ(K) = 1 for K ∈ Kn.

Proof. The proof uses induction with respect to the dimension. For n = 0,the existence is trivial. Suppose that n > 0 and the existence has beenproved in Euclidean spaces of dimension n − 1. We choose a unit vectoru ∈ Rn and define

χ(K) :=∑λ∈R

[χ(K ∩H(u, λ))− lim

µ↓λχ(K ∩H(u, µ))

](19)

for K ∈ U(Kn). On the right-hand side, χ denotes the additive functionthat exists by the induction hypothesis in spaces of dimension n − 1. It isobvious that χ(K) = 1 for K ∈ Kn. If K = K1 ∪ · · · ∪Km with Ki ∈ Kn,then the inclusion-exclusion principle gives

χ(K ∩H(u, λ)) =∑

v∈S(m)

(−1)|v|−1χ(Kv ∩H(u, λ)),

25

since χ is additive on the polyconvex sets in H(u, λ). Now the functionλ 7→ χ(Kv ∩H(u, λ)) is the indicator function of a compact interval, henceit is clear that the limit in (19) exists for every λ ∈ R and is non-zero only forfinitely many values of λ. Thus χ is well-defined on U(Kn). It follows from(19) and the induction hypothesis that χ is additive on U(Kn). This provesthe existence of χ. The uniqueness is clear from the inclusion-exclusionprinciple.

The function χ is called the Euler characteristic. It coincides, on U(Kn),with the Euler characteristic as defined in algebraic topology.

Another simple example of a valuation on U(Kn) is given by the indicatorfunction. For K ∈ U(Kn), let

1K(x) :=

1 for x ∈ K,

0 for x ∈ Rn \K.

For K,L ∈ U(Kn) we trivially have

1K∪L(x) + 1K∩L(x) = 1K(x) + 1L(x)

for x ∈ Rn. Hence, the mapping

ϕ : U(Kn) → V

K 7→ 1K

is an additive function on U(Kn) with values in the vector space V of finitelinear combinations of indicator functions of polyconvex sets. SinceK 7→ 1K

is additive, for K ∈ U(Kn) with K = K1∪· · ·∪Km, Ki ∈ Kn, the inclusion-exclusion principle gives

1K =∑

v∈S(m)

(−1)|v|−11Kv .

Thus V consists of all linear combinations of indicator functions of convexbodies.

We will now prove a general extension theorem for valuations on Kn,which is due to Groemer [1]. We endow the set Kn of convex bodies withthe Hausdorff metric δ, which is defined by

δ(K,L) := maxmaxx∈K

miny∈L

‖x− y‖, maxx∈L

miny∈K

‖x− y‖

= minε > 0 : K ⊂ L+ εBn, L ⊂ K + εBn,

26

and with the induced topology. A general extension theorem holds for con-tinuous valuations with values in a topological vector space. This theoremwill imply Theorem 4.1, but the short proof of the latter is of independentinterest.

4.2 Theorem. Let X be a topological vector space, and let ϕ : Kn → Xbe a continuous additive mapping. Then ϕ has an additive extension to theconvex ring U(Kn).

Proof. An essential part of the proof is the following

Proposition. The equalitym∑

i=1

αi1Ki= 0

with m ∈ N, αi ∈ R, Ki ∈ Kn impliesm∑

i=1

αiϕ(Ki) = 0.

Assume this proposition were false. Then there is a smallest number m ∈ N,necessarilym ≥ 2, for which there exist numbers α1, . . . , αm ∈ R and convexbodies K1, . . . ,Km ∈ Kn such that

m∑i=1

αi1Ki= 0, (20)

butm∑

i=1

αiϕ(Ki) =: a 6= 0. (21)

Let H ⊂ Rn be a hyperplane with K1 ⊂ intH+, where H+,H− are the twoclosed halfspaces bounded by H. By (20) we have

m∑i=1

αi1Ki∩H− = 0,m∑

i=1

αi1Ki∩H = 0.

Since K1 ∩H− = ∅ and K1 ∩H = ∅, each of these two sums has at mostm−1 non-zero summands. From the mimimality of m (and from ϕ(∅) = 0)we get

m∑i=1

αiϕ(Ki ∩H−) = 0,m∑

i=1

αiϕ(Ki ∩H) = 0.

27

The additivity of ϕ on Kn yields

m∑i=1

αiϕ(Ki ∩H+) = a, (22)

whereas (20) givesm∑

i=1

αi1Ki∩H+ = 0. (23)

A standard separation theorem for convex bodies implies the existence of asequence (Hj)j∈N of hyperplanes with K1 ⊂ intH+

j for j ∈ N and

K1 =∞⋂

j=1

H+j .

If the argument that has led us from (20), (21) to (23), (22) is appliedk-times, we obtain

m∑i=1

αiϕ

Ki ∩k⋂

j=1

H+j

= a.

For k →∞ this yields

m∑i=1

αiϕ(Ki ∩K1) = a, (24)

since

limk→∞

Ki ∩k⋂

j=1

H+j = Ki ∩K1

in the sense of the Hausdorff metric (if Ki∩K1 6= ∅, otherwise use ϕ(∅) = 0)and ϕ is continuous. Equality (20) implies

m∑i=1

αi1Ki∩K1 = 0. (25)

The procedure leading from (20) and (21) to (25) and (24) can berepeated, replacing the bodies Ki and K1 by Ki ∩ K1 and K2, then byKi ∩K1 ∩K2 and K3, and so on. Finally one obtains

m∑i=1

αi1K1∩···∩Km= 0

28

andm∑

i=1

αiϕ(K1 ∩ · · · ∩Km) = a

(because of Ki ∩ K1 ∩ · · · ∩ Km = K1 ∩ · · · ∩ Km). Now a 6= 0 implies∑mi=1 αi 6= 0 and hence 1K1∩···∩Km

= 0 by the first relation, but this yieldsϕ(K1 ∩ · · · ∩ Km) = 0, contradicting the second relation. This completesthe proof of the proposition.Now we consider the real vector space V of all finite linear combinations ofindicator functions of elements of Kn. For K ∈ U(Kn) we have 1K ∈ V , asnoted earlier. For fixed f ∈ V we choose a representation

f =m∑

i=1

αi1Ki

with m ∈ N, αi ∈ R, Ki ∈ Kn and define

ϕ(f) :=m∑

i=1

αiϕ(Ki).

The proposition proved above shows that this definition is possible, sincethe right-hand side does not depend on the special representation chosenfor f . Evidently, ϕ : V → X is a linear map satisfying ϕ(1K) = ϕ(K) forK ∈ Kn. We can now extend ϕ from Kn to U(Kn) by defining

ϕ(K) := ϕ(1K) for K ∈ U(Kn).

By the linearity of ϕ and the additivity of the map K 7→ 1K we obtain, forK,M ∈ U(Kn),

ϕ(K ∪M) + ϕ(K ∩M) = ϕ(1K∪M ) + ϕ(1K∩M )= ϕ(1K∪M + 1K∩M )= ϕ(1K + 1M )= ϕ(1K) + ϕ(1M )= ϕ(K) + ϕ(M).

Thus ϕ is additive on U(Kn).

5 Local parallel sets and curvature measures

One of our aims will be to compute integrals such as

I(K,M) :=∫

Gn

χ(K ∩ gM) dµ(g) (26)

29

for convex bodies K,M ∈ Kn, where µ is the invariant measure on themotion group Gn; thus I(K,M) is the total Haar measure of the set of rigidmotions which bring M into a hitting position with K. We get a first hintto what the result will involve if we choose for M a ball ρBn of radius ρ > 0.In that case,

I(K, ρBn) =∫

Rn

χ(K ∩ (ρBn + t)) dλ(t) = Vn(K + ρBn),

as obtained in Section 2. The set K + ρBn is known as the outer parallelset of K at distance ρ. It can also be represented as

K + ρBn = x ∈ Rn : d(K,x) ≤ ρ,

whered(K,x) := min‖x− y‖ : y ∈ K

is the distance of x from K. A fundamental result in the geometry ofconvex bodies, the Steiner formula, says that the volume Vn(K + ρBn) ofthe parallel body, as a function of the parameter ρ, is a polynomial of degreen, thus

Vn(K + ρBn) =n∑

i=0

ρn−iκn−iVi(K). (27)

The reason for introducing the normalizing factors κn−i will become clearlater in this section. The coefficients V0(K), . . . , Vn(K) appearing in (27)define important functionals of K. We have just seen that they inevitablyappear when we want to compute the integral I(K, ρBn). As it turns out,also the general integral I(K,M) given by (26) can be expressed in termsof these functionals alone, evaluated for the bodies K and M .

In the present section, a more general version of the Steiner formula(27) will be obtained. Namely, we replace the parallel body K + ρBn by alocal version of it. The polynomial expansion generalizing (27) then definesa series of measures on Rn, the curvature measures of the convex body K.These measures will appear in very general versions of the kinematic formulaof integral geometry.

We need a simple device from convex geometry. Let K ∈ Kn be a convexbody. For x ∈ Rn, there is a unique point p(K,x) in K nearest to x, thatis,

‖p(K,x)− x‖ = min‖y − x‖ : y ∈ K = d(K,x).

30

This defines a continuous map p(K, ·) : Rn → K, which is called the nearest-point map of K, or the metric projection onto K. Also the map

p : Kn × Rn → Rn,

(K,x) 7→ p(K,x)

is continuous.Now for K ∈ Kn, a Borel set A ∈ B(Rn) and a number ρ ≥ 0, we define

the local parallel set of (K,A) at distance ρ by

Mρ(K,A) := x ∈ Rn : d(K,x) ≤ ρ, p(K,x) ∈ A.

This is a Borel set, since p(K, ·) is continuous. We can, therefore, define

µρ(K,A) := λ(Mρ(K,A)) for A ∈ B(Rn).

In other words, µρ(K, ·) is the image measure of the Lebesgue measure,restricted to the parallel body Kρ = K+ρBn, under the nearest point mapof K. In particular, µρ(K, ·) is a finite measure on B(Rn). We call it thelocal parallel volume of K at distance ρ.

This measure is concentrated on K, that is, µρ(K,A) = µρ(K,A ∩K).We first prove some fundamental properties of the mapping µρ : Kn ×

B(Rn) → R. In the following, w→ denotes weak convergence of finite mea-sures.

5.1 Theorem. Let (Kj)j∈N be a sequence in Kn satisfying Kj → K forj →∞. Then

µρ(Kj , ·)w→ µρ(K, ·) for j →∞, (28)

for every ρ > 0.

Proof. By a well-known characterization of weak convergence, we have toshow that

lim infj→∞

µρ(Kj , A) ≥ µρ(K,A) (29)

for every open set A, and

limj→∞

µρ(Kj ,Rn) = µρ(K,Rn). (30)

Let A ⊂ Rn be open. Let x ∈ Mρ(K,A) be a point with d(K,x) < ρ.Since p is continuous, we have p(Kj , x) → p(K,x) and d(Kj , x) → d(K,x)for j → ∞. Hence, for all sufficiently large j we deduce that p(Kj , x) ∈ Aand d(Kj , x) < ρ, hence x ∈Mρ(Kj , A). Thus we have

Mρ(K,A) \ ∂Kρ ⊂ lim infj→∞

Mρ(Kj , A)

31

and, therefore,

µρ(K,A) = λ(Mρ(K,A) \ ∂Kρ)

≤ λ

(lim infj→∞

Mρ(Kj , A))

≤ lim infj→∞

λ(Mρ(Kj , A))

= lim infj→∞

µρ(Kj , A),

which proves (29). The relation (30) follows from standard results of convexgeometry.

5.2 Theorem. For any Borel set A ∈ B(Rn) and any ρ > 0, the functionµρ(·, A) : Kn → R is measurable.

Proof. For an open set A, the preceding proof shows that the functionµρ(·, A) is lower semicontinuous, hence it is measurable.

Denote by A the system of all sets A ∈ B(Rn) for which µρ(·, A) ismeasurable. We show that A is a Dynkin system. For A1, A2 ∈ A withA2 ⊂ A1 we have Mρ(K,A2) ⊂Mρ(K,A1) and

Mρ(K,A1 \A2) = Mρ(K,A1) \Mρ(K,A2),

henceµρ(K,A1 \A2) = µρ(K,A1)− µρ(K,A2)

for all K ∈ Kn, which shows that A1 \ A2 ∈ A. If (Aj)j∈N is a disjointsequence in A, then

µρ

K, ∞⋃j=1

Aj

=∞∑

j=1

µρ(K,Aj)

for K ∈ Kn, since µρ(K, ·) is a measure. It follows that⋃∞

j=1Aj ∈ A. ThusA is a Dynkin system. Since it contains the open sets, it also contains theσ-algebra generated by the open sets and thus all Borel sets, as asserted.

5.3 Theorem. For any Borel set A ∈ B(Rn) and for ρ > 0, the functionµρ(·, A) : Kn → R is additive.

Proof. Let K,L ∈ Kn be convex bodies with K ∪ L ∈ Kn. Let x ∈ Rn, andput y := p(K ∪ L, x). We assume y ∈ K, without loss of generality. Then

p(K ∪ L, x) = p(K,x). (31)

32

Let z := p(L, x). Since K ∪ L is convex, there is a point a ∈ [z, y] (thesegment with endpoints z and y) with a ∈ K ∩ L. From y = p(K ∪ L, x) itfollows that ‖y − x‖ ≤ ‖z − x‖ and hence ‖a − x‖ ≤ ‖z − x‖. From a ∈ Land the definition of z we conclude that a = z and thus z ∈ K ∩ L. Thisshows that

p(K ∩ L, x) = p(L, x). (32)

For K ′ ∈ Kn, let 1ρ(K ′, A, ·) be the indicator function of the local parallelset Mρ(K ′, A). From (31) and (32) it follows that

1ρ(K ∪ L,A, x) = 1ρ(K,A, x), 1ρ(K ∩ L,A, x) = 1ρ(L,A, x).

Since x was arbitrary, this yields

1ρ(K ∪ L,A, ·) + 1ρ(K ∩ L,A, ·) = 1ρ(K,A, ·) + 1ρ(L,A, ·).

Integrating this equation with the Lebesgue measure, we obtain

µρ(K ∪ L,A) + µρ(K ∩ L,A) = µρ(K,A) + µρ(L,A),

which shows that µρ(·, A) is additive on Kn.

We will now explicitly compute the local parallel volume in the case of aconvex polytope. For this, we need some elementary facts about polytopes,which we will use without proof.

A polyhedral set in Rn is a set which can be represented as the intersec-tion of finitely many closed halfspaces. A bounded non-empty polyhedralset is called a convex polytope or briefly a polytope. Let P be a polytope.If H is a supporting hyperplane of P , then P ∩ H is again a polytope.The set F := P ∩ H is called a face of P , and an m-face if dim F = m,m ∈ 0, . . . , n− 1. If dim P = n, we consider P as an n-face of itself. ByFm(P ) we denote the set of all m-faces of P . For F ∈ Fm(P ) we define

λF (B) := λ(m)(B ∩ F ) for B ∈ B(Rn),

where λ(m) denotes m-dimensional Lebesgue measure. For F ∈ Fm(P ),m ∈ 0, . . . , n− 1 and a point x ∈ relintF (the relative interior of F ), letN(P, F ) be the normal cone of P at F ; this is the cone of outer normalvectors of supporting hyperplanes to P at x. It does not depend upon thechoice of x. The number

γ(F, P ) :=λ(n−m)(N(P, F ) ∩Bn)

κn−m

is called the external angle of P at its face F . We also put γ(P, P ) = 1 andγ(F, P ) = 0 if either F = ∅ or F is not a face of P .

33

Now let a polytope P , a Borel set A ∈ B(Rn) and a number ρ > 0 begiven. For x ∈ Rn, the nearest point p(P, x) lies in the relative interior of aunique face of P . Therefore,

Mρ(P,A) =n⋃

m=0

⋃F∈Fm(P )

[Pρ ∩ p(P, ·)−1(A ∩ relintF )

](33)

is a disjoint decomposition of the local parallel set Mρ(P,A). It follows fromthe properties of the nearest point map that

Pρ ∩ p(P, ·)−1(A ∩ relintF ) (34)= (A ∩ relintF )⊕ (N(P, F ) ∩ ρBn), (35)

where ⊕ denotes direct sum. An application of Fubini’s theorem gives

λ(Pρ ∩ p(P, ·)−1(A ∩ relintF ))

= λ(m)(A ∩ F )λ(n−m)(N(P, F ) ∩ ρBn)= λ(m)(A ∩ F )ρn−mκn−mγ(F, P ).

Together with (33), this yields

µρ(P,A) =n∑

m=0

ρn−mκn−m

∑F∈Fm(P )

λ(m)(A ∩ F )γ(F, P ).

Hence, if we define a measure Φm(P, ·) on B(Rn) by

Φm(P, ·) :=∑

F∈Fm(P )

γ(F, P )λF ,

then

µρ(P,A) =n∑

m=0

ρn−mκn−mΦm(P,A).

This gives the desired polynomial expansion of the local parallel volume inthe case of polytopes. The following theorem extends this result to generalconvex bodies.

5.4 Theorem. (Local Steiner formula) For every convex body K ∈ Kn,there exist finite measures Φ0(K, ·), . . . ,Φn(K, ·) on B(Rn) such that thelocal parallel volume satisfies

µρ(K,A) =n∑

j=0

ρn−jκn−jΦj(K,A)

34

for all A ∈ B(Rn) and all ρ ≥ 0.

Proof. If P is a polytope, we have seen above that

µρ(P,A) =n∑

j=0

ρn−jκn−jΦj(P,A) (36)

withΦj(P, ·) =

∑F∈Fj(P )

γ(F, P )λF . (37)

Now let K ∈ Kn be an arbitrary convex body. As one knows from convexgeometry, there is a sequence (Pi)i∈N of polytopes converging to K in theHausdorff metric. In (36), we replace P by Pi and ρ by each of the numbers1, . . . , n+ 1. The resulting system of linear equations,

µk(Pi, A) =n∑

j=0

kn−jκn−jΦj(Pi, A), k = 1, . . . , n+ 1,

can be solved for the ‘unknowns’ κn−jΦj(Pi, A) (it has a Vandermondedeterminant), which yields representations

Φj(Pi, A) =n+1∑k=1

αjkµk(Pi, A), j = 0, . . . , n.

Here the coefficients αjk do not depend on Pi or A, thus we have

Φj(Pi, ·) =n+1∑k=1

αjkµk(Pi, ·) for i ∈ N.

By Theorem 5.1, for each fixed ρ ≥ 0 the measures µρ(Pi, ·) converge weaklyto µρ(K, ·). Hence, if we define a finite signed measure by

Φj(K, ·) :=n+1∑k=1

αjkµk(K, ·),

then the measures Φj(Pi, ·) converge, for i → ∞, weakly to the signedmeasure Φj(K, ·) (j = 0, . . . , n). It follows that the latter is nonnegative,and it also follows that

µρ(K, ·) =n∑

j=0

ρn−jκn−jΦj(K, ·),

35

using (36) and weak convergence.

One calls Φj(K, ·) the jth curvature measure of the body K ∈ Kn. Thereason for this name becomes clear if one considers a convex body K whoseboundary is a regular hypersurface of class C2. In that case, the localparallel volume can be computed by differential-geometric means, and oneobtains for j = 0, . . . , n− 1 the representation

Φj(K,A) =

(nj

)nκn−j

∫A∩∂K

Hn−1−j dS.

Here Hk denotes the kth normalized elementary symmetric function of theprincipal curvatures of ∂K, and dS is the volume form on ∂K. Thus thecurvature measures are (up to constant factors) indefinite integrals of cur-vature functions, and they replace the latter in the non-smooth case.

For j = n, we simply have

Φn(K,A) = λ(K ∩A) for A ∈ B(Rn),

as follows immediately from the definition of the local parallel set and thelocal Steiner formula. For a general convex body K it is clear that themeasures Φ0(K, ·), . . . ,Φn−1(K, ·) are concentrated on ∂K, since µρ(K,A)−λ(K ∩A) depends only on A ∩ ∂K.

For polytopes P , we have the explicit representation (37) of the curva-ture measures. The external angle appearing in it does not depend on thedimension of the surrounding space, as follows easily from Fubini’s theorem.In other words, if dimP < n, it makes no difference if the external angleγ(F, P ) is computed in Rn or in the affine hull of P . This independenceof dimension extends to the curvature measures Φj(P, ·) and then, by ap-proximation and weak convergence, to the curvature measures Φj(K, ·) ofarbitrary convex bodies.

We mention without proof that for arbitrary convex bodies K the mea-sures Φ0(K, ·) and Φn−1(K, ·) have simple intuitive interpretations. Namely,if dimK 6= n− 1, then

Φn−1(K,A) =12Hn−1(A ∩ ∂K).

For dimK = n − 1, one trivially has Φn−1(K,A) = Hn−1(A ∩ ∂K). Themeasure Φ0 is the normalized area of the spherical image. Let σ(K,A) ⊂Sn−1 denote the set of all outer unit normal vectors ofK at points of A∩∂K,then

Φ0(K,A) =1nκn

Hn−1(σ(K,A)).

36

We can use the relation

Φj(K, ·) =n+1∑k=1

αjkµk(K, ·), (38)

which was obtained in the proof of Theorem 5.4, to transfer properties of thelocal parallel volumes µρ(K, ·) to the curvature measures Φj(K, ·). In thisway Theorems 5.1, 5.2, 5.3, together with some easily obtained additionalproperties of the local parallel volumes, yield a series of properties of thecurvature measures, which we list in the following theorem.

5.5 Theorem. Let j ∈ 0, . . . , n.(a) Φj(K, ·) depends weakly continuously on K, that is, Ki → K implies theweak convergence Φj(Ki, ·)

w→ Φj(K, ·) for i→∞.(b) For every A ∈ B(Rn), the function Φj(·, A) is measurable on Kn.(c) Φj is motion covariant, that is,

Φj(gK, gA) = Φj(K,A)

for every rigid motion g ∈ Gn and all K ∈ Kn and A ∈ B(Rn).(d) Φj is homogeneous of degree j, that is,

Φj(αK,αA) = αjΦ(K,A)

for every α > 0 and all K ∈ Kn and A ∈ B(Rn).(e) Φj is defined locally, that is, for every open set A ⊂ Rn and all convexbodies K,M ∈ Kn with K ∩A = M ∩A, one has

Φj(K,B) = Φj(M,B)

for every Borel set B ⊂ A.(f) Φj(·, A) is additive for every A ∈ B(Rn), that is,

Φj(K ∪ L,A) + Φj(K ∩ L,A) = Φj(K,A) + Φj(L,A)

holds for all convex bodies K,L ∈ Kn with K ∪ L ∈ Kn.

The final property, together with Theorem 4.2, has the important conse-quence that the curvature measures have an additive extension to polycon-vex sets. This means that one can define signed measures on the convex ringU(Kn) in the following way. Let K ∈ U(Kn) and choose a representationK =

⋃mi=1Ki with m ∈ N and Ki ∈ Kn. Then

Φj(K, ·) :=∑

v∈S(m)

(−1)|v|−1Φj(Kv, ·)

37

does not depend on the special choice of the representation; in particular,this is consistent with the already defined value Φj(K, ·) for convex K.This follows from Theorem 4.2, since the weak convergence of curvaturemeasures can be interpreted as convergence in the vector space of finitesigned measures, on a suitable compact subset of Rn, with respect to asuitable topology.

We have now everything at hand to formulate a central result of integralgeometry. This is the principal kinematic formula, in a version for curvaturemeasures on the convex ring. Let K,M ∈ U(Kn) be polyconvex sets, letA,B ∈ B(Rn) be Borel sets, and let j ∈ 0, . . . , n. Then∫

Gn

Φj(K ∩ gM,A ∩ gB) dµ(g) =n∑

k=j

αnjkΦj(K,A)Φn+j−k(M,B)

holds, with certain explicit constants αnjk.We will indicate a proof of this result in Section 9. Before that, however,

we will prove a global version of this formula in a different way. The methodof proof is of independent interest and leads to further results for which noother access is known.

The global result refers to the total measures

Vj(K) := Φj(K,Rn), j = 0, . . . , n.

The number Vj is called the jth intrinsic volume of K. These importantfunctionals are defined by the classical Steiner formula

Vn(K + ρBn) =n∑

j=0

ρn−jκn−jVj(K),

of which Theorem 5.4 is the local generalization. As a function on Kn, eachintrinsic volume Vj is continuous, additive and rigid motion invariant. Inthe next section we shall prove that the intrinsic volumes are essentiallycharacterized by these properties.

The additive extensions of the intrinsic volumes to the convex ringU(Kn) will be denoted by the same symbols. In the following cases, theyhave simple intuitive interpretations. It is clear that

Vn(K) = λ(K) for K ∈ U(Kn),

since this holds true for convex bodies K and both functions, Vn and λ, areadditive on U(Kn). It also remains true for polyconvex sets that

2Vn−1(K) = Hn−1(∂K)

38

if K is the closure of its interior, but this requires an extra proof. Finally,

V0(K) = χ(K) for K ∈ U(Kn),

so that V0 is nothing but the Euler characteristic. For a convex polytope Pwe have

V0(P ) = Φ0(P,Rn) =∑

F∈F0(P )

γ(F, P ) = 1,

since the normal cones N(P, F ) of P at its vertices F cover Rn and havepairwise no common interior points. By additivity, the equation V0(K) =χ(K) extends from Kn to U(Kn).

6 Hadwiger’s characterization theorem

The jth intrinsic volume Vj : Kn → R is an additive, continuous and rigidmotion invariant function. A celebrated theorem due to Hadwiger (see [2])says that any function on Kn with these properties is a linear combinationof the intrinsic volumes V0, . . . , Vn. This result can be used to prove someformulae of the integral geometry of convex bodies in a very elegant way.Whereas Hadwiger’s original proof was quite long, one has now a shorterproof due to Klain [3]. We will present his proof here, except that at onepoint we take a certain analytical result for granted.

The crucial step for a proof of the characterization theorem is the fol-lowing result.

6.1 Theorem. Suppose that ψ : Kn → R is an additive, continuous, motioninvariant function satisfying ψ(K) = 0 whenever either dimK < n or K isa unit cube. Then ψ = 0.

Proof. The proof proceeds by induction with respect to the dimension. Forn = 0, there is nothing to prove. If n = 1, ψ vanishes on (closed) segmentsof unit length, hence on segments of length 1/k for k ∈ N and therefore onsegments of rational length. By continuity, ψ vanishes on all segments andthus on K1.

Now let n > 1 and suppose that the assertion has been proved in di-mensions less than n. Let H ⊂ Rn be a hyperplane and I a closed seg-ment of length 1, orthogonal to H. For convex bodies K ⊂ H defineϕ(K) := ψ(K + I). Clearly ϕ has, relative to H, the properties of ψ in theTheorem, hence the induction hypothesis yields ϕ = 0. For fixed K ⊂ H,we thus have ψ(K+ I) = 0, and a similar argument as used above for n = 1

39

shows that ψ(K + S) = 0 for any closed segment S orthogonal to H. Thusµ vanishes on right convex cylinders.

Let K ⊂ H again be a convex body and let S = conv 0, s be a segmentnot parallel to H. If m ∈ N is sufficiently large, the cylinder Z := K +mScan be cut by a hyperplane H ′ orthogonal to S so that the two closedhalfspaces H−,H+ bounded by H ′ satisfy K ⊂ H− and K + ms ⊂ H+.Then Z := [(Z ∩H−) +ms] ∪ (Z ∩H+) is a right cylinder, and we deducethat mµ(K + S) = µ(Z) = µ(Z) = 0. Thus ψ vanishes on arbitrary convexcylinders.

By Theorem 4.2, the continuous additive function ψ has an additiveextension to the convex ring; this extension is also denoted by ψ. It followsthat

ψ

(k⋃

i=1

Ki

)=

k∑i=1

ψ(Ki)

whenever K1, . . . ,Kk are convex bodies such that dim (Ki ∩ Kj) < n fori 6= j.

Let P be a polytope and S a segment. The sum P + S has a decompo-sition P +S =

⋃ki=1 Pi, where P1 = P , the polytope Pi is a convex cylinder

for i > 1, and dim (Pi ∩Pj) < n for i 6= j. It follows that ψ(P +S) = ψ(P ).By induction, we obtain ψ(P +Z) = ψ(P ) if Z is a finite sum of segments.By continuity, ψ(K+Z) = ψ(K) for arbitrary convex bodies K and zonoidsZ, that is, limits of sums of segments.

Now we have to use an analytic result, for which we do not give a proof.Let K be a centrally symmetric convex body which is sufficiently smooth(say, its support function is of class C∞). Then there exist zonoids Z1, Z2

so that K + Z1 = Z2 (this can be seem from Section 3.5 in [7], especiallyTheorem 3.5.3). We conclude that ψ(K) = ψ(K + Z1) = ψ(Z2) = 0. Sinceevery centrally symmetric convex body K can be approximated by bodieswhich are centrally symmetric and sufficiently smooth in the above sense, itfollows from the continuity of ψ that ψ(K) = 0 for all centrally symmetricconvex bodies.

Now let ∆ be a simplex, say ∆ = conv 0, v1, . . . , vn, without loss ofgenerality. Let v := v1 + · · · + vn and ∆′ := conv v, v − v1, . . . , v − vn,then ∆′ = −∆ + v. The vectors v1, . . . , vn span a parallelotope P . It is theunion of ∆,∆′ and the part of P lying between the hyperplanes spannedby v1, . . . , vn and v − v1, . . . , v − vn, respectively. The latter, say Q, is acentrally symmetric polytope, and ∆ ∩ Q, ∆′ ∩ Q are of dimension n − 1.We deduce that 0 = ψ(P ) = ψ(∆) + ψ(Q) + ψ(∆′), thus ψ(−∆) = −ψ(∆).If the dimension n is even, then −∆ is obtained from ∆ by a proper rigidmotion, and the motion invariance of ψ yields ψ(∆) = 0. If the dimension

40

n > 1 is odd, we decompose ∆ as follows. Let z be the centre of theinscribed ball of ∆, and let pi be the point where this ball touches the facetFi of ∆ (i = 1, . . . , n + 1). For i 6= j, let Qij be the convex hull of theface Fi ∩ Fj and the points z, pi, pj . The polytope Qij is invariant underreflection in the hyperplane spanned by Fi ∩ Fj and z. If Q1, . . . , Qm arethe polytopes Qij for 1 ≤ c < j ≤ n + 1 in any order, then P =

⋃mr=1Qr

and dim (Qr ∩ Qs) < n for r 6= s. Since −Qr is the image of Qr undera proper rigid motion, we have ψ(−∆) =

∑ψ(−Qr) =

∑ψ(Qr) = ψ(∆).

Thus ψ(∆) = 0 for every simplex ∆.Decomposing a polytope P into simplices, we obtain ψ(P ) = 0. The

continuity of ψ now implies ψ(K) = 0 for all convex bodies K. This finishesthe induction and hence the proof of Theorem 6.1.

Hadwiger’s characterization theorem is now an easy consequence.

6.2 Theorem. Suppose that ψ : Kn → R is an additive, continuous, motioninvariant function. Then there are constants c0, . . . , cn so that

ψ(K) =n∑

i=0

ciVi(K)

for all K ∈ Kn.

Proof. We use induction on the dimension. For n = 0 the assertion is trivial.Suppose that n > 0 and the assertion has been proved in dimensions lessthan n. Let H ⊂ Rn be a hyperplane. The restriction of ψ to the convexbodies lying in H is additive, continuous and invariant under motions ofH into itself. By the induction hypothesis, there are constants c0, . . . , cn−1

so that ψ(K) =∑n−1

i=0 ciVi(K) holds for convex bodies K ⊂ H (note thatthe intrinsic volumes do not depend on the dimension of the surroundingspace). By the motion invariance of ψ and Vi, this holds for all K ∈ Kn ofdimension less than n. It follows that the function ψ′ defined by

ψ′(K) := ψ(K)−n∑

i=0

ciVi(K)

for K ∈ Kn, where cn is chosen so that ψ′ vanishes at a fixed unit cube,satisfies the assumptions of Theorem 6.1. Hence ψ′ = 0, which completesthe proof of Theorem 6.2.

The late Gian-Carlo Rota, in a Colloquium Lecture at the Annual Meetingof the AMS in 1997, called Hadwiger’s characterization theorem the ‘Main

41

Theorem of Geometric Probability’. The reason is that it can be usedto derive kinematic formulae of integral geometry, which can in turn beinterpreted in terms of geometric hitting probabilities. We shall see this, inmore elaborate versions, in the next two sections.

7 Kinematic and Crofton formulae

Our aim in this section will be to compute the integrals∫Gn

Vj(K ∩ gM) dµ(g)

and ∫En

k

Vj(K ∩ E) dµk(E)

for convex bodies K,M , where Vj is an intrinsic volume. For that, we useHadwiger’s characterization theorem. From this result, we first deduce amore general kinematic formula, involving a functional on convex bodiesthat need not have any invariance property.

7.1 Theorem. If ϕ : Kn → R is an additive continuous function, then∫Gn

ϕ(K ∩ gM) dµ(g) =n∑

k=0

ϕn−k(K)Vk(M) (39)

for K,M ∈ Kn, where the coefficients ϕn−k(K) are given by

ϕn−k(K) =∫En

k

ϕ(K ∩ E) dµk(E). (40)

Proof. In order that the integral in (39) makes sense, we first have toshow that for given convex bodies K,M the function g 7→ ϕ(K ∩ gM) isµ-integrable. Let Gn(K,M) denote the set of all rigid motions g ∈ Gn

for which K and gM touch, that is, K ∩ gM 6= ∅ but K and gM can beseparated weakly by a hyperplane. Using the map γ from (8), it is easy tosee that γ(t, ϑ) ∈ Gn(K,M) holds if and only if t ∈ ∂(K − ϑM); hence

µ(Gn(K,M)) =∫

SOn

∫Rn

1Gn(K,M)(γ(t, ϑ)) dλ(t) dν(ϑ)

42

=∫

SOn

λ(∂(K − ϑM)) dν(ϑ) = 0.

On Gn \ Gn(K,M), the map g 7→ ϕ(K ∩ gM) is continuous. Since thecontinuous function ϕ is bounded on the compact set K ′ ∈ Kn : K ′ ⊂ K,it follows that the integral in (39) is well-defined and finite.

Now we fix a convex body K ∈ Kn and define

ψ(M) :=∫

Gn

ϕ(K ∩ gM) dµ(g) for M ∈ Kn.

Then ψ : Kn → R is obviously additive and motion invariant. It followsfrom the bounded convergence theorem that ψ is continuous. Theorem 6.2yields the existence of constants ϕ0(K), . . . , ϕn(K) so that

ψ(M) =n∑

k=0

ϕk(K)Vn−k(M)

for all M ∈ Kn. The constants depend, of course, on the given body K,and we have now to determine them.

Suppose first that 1 ≤ k ≤ n − 1 and let Lk ∈ Lnk . We choose a k-

dimensional cube W ⊂ Lk with 0 ∈ W and λ(k)(W ) = 1. For r ≥ 1 wehave

ψ(rW ) =∫

Gn

ϕ(K ∩ grW ) dµ(g) =n∑

i=0

ϕn−i(K)Vi(rW ).

The intrinsic volumes have the easily established properties

Vi(rW ) =

0 for i > k,

rk for i = k,

riVi(W ) for i < k.

This yieldsψ(rW ) = ϕn−k(K)rk +O(rk−1) (41)

for r →∞. On the other hand,

ψ(rW ) =∫

Gn

ϕ(K ∩ grW ) dµ(g)

=∫

SOn

∫Rn

ϕ(K ∩ (ϑrW + x)) dλ(x) dν(ϑ)

43

=∫

SOn

∫ϑL⊥k

∫ϑLk

ϕ(K ∩ (ϑrW + x1 + x2)) dλ(k)(x2)

dλ(n−k)(x1) dν(ϑ).

For fixed ϑ ∈ SOn and x1 ∈ L⊥k we put

X := x2 ∈ ϑLk : K ∩ (ϑrW + x1 + x2) = K ∩ (ϑLk + x1),Y := x2 ∈ ϑLk : ∅ 6= K ∩ (ϑrW + x1 + x2) 6= K ∩ (ϑLk + x1).

Then ∫ϑLk

ϕ(K ∩ (ϑrW + x1 + x2)) dλ(k)(x2)

= ϕ(K ∩ (ϑLk + x1))∫X

dλ(k)(x2)

+∫Y

ϕ(K ∩ (ϑrW + x1 + x2)) dλ(k)(x2).

For r →∞, we get ∫X

dλ(k)(x2) = rk +O(rk−1).

Since ϕ is bounded on compact sets,∫Y

ϕ(K ∩ (ϑrW + x1 + x2)) dλ(k)(x2) = O(rk−1).

We deduce that

ψ(rW ) = rk

∫SOn

∫ϑL⊥k

ϕ(K ∩ (ϑLk + x1)) dλ(n−k)(x1) dν(ϑ) +O(rk−1)

= rk

∫SOn

∫L⊥k

ϕ(K ∩ ϑ(Lk + x1)) dλ(n−k)(x1) dν(ϑ) +O(rk−1)

= rk

∫En

k

ϕ(K ∩ E) dµk(E) +O(rk−1).

If we compare this with (41) and let r tend to infinity, we obtain the assertedformula (40) for the coefficients.

44

In the cases k = 0 and k = n, simpler versions of the proof, with theobvious changes, give the same result. This completes the proof of Theorem7.1.

In Theorem 7.1, we can choose for ϕ the intrinsic volume Vj and get∫Gn

Vj(K ∩ gM) dµ(g) =n∑

k=0

Vj,n−k(K)Vk(M)

withVj,n−k(K) =

∫En

k

Vj(K ∩ E) dµk(E).

By

ψ(K) :=∫En

k

Vj(K ∩ E) dµk(E) for K ∈ Kn

we again define a functional ψ : Kn → R which is additive, continuous andmotion invariant. This is proved similarly as above. Hadwiger’s characteri-zation theorem yields a representation

ψ(K) =n∑

r=0

crVr(K).

Here only one coefficient is non-zero. In fact, from

ψ(K) =∫Ln

k

∫L⊥

Vj(K ∩ (L+ y)) dλ(n−k)(y) dνk(L)

one sees that ψ has the homogeneity property

ψ(αK) = αn−k+jψ(K)

for α > 0. Since Vk is homogeneous of degree k, we deduce that cr = 0 forr 6= n− k + j. Thus we have obtained∫

Enk

Vj(K ∩ E) dµk(E) = αnjkVn+j−k(K)

with some constant αnjk. In order to determine this constant, we choosefor K the unit ball Bn. For ε ≥ 0, the Steiner formula gives

n∑j=0

εn−jκn−jVj(Bn) = Vn(Bn + εBn) = (1 + ε)nκn =n∑

j=0

εn−j

(n

j

)κn,

45

hence

Vj(Bn) =

(nj

)κn

κn−jfor j = 0, . . . , n.

Choosing L ∈ Lnk , we obtain

αnjkVn+j−k(Bn) =∫En

k

Vj(Bn ∩ E) dµk(E)

=∫

SOn

∫L⊥

Vj(Bn ∩ ϑ(L+ x)) dλ(n−k)(x) dν(ϑ)

=∫

L⊥∩Bn

(1− ‖x‖2)j/2Vj(Bn ∩ L) dλ(n−k)(x)

=

(kj

)κk

κk−j

∫L⊥∩Bn

(1− ‖x‖2)j/2 dλ(n−k)(x).

Introducing polar coordinates, the latter integral is transformed into a Betaintegral, and one obtains∫

L⊥∩Bn

(1− ‖x‖2)j/2 dλ(n−k)(x)

= (n− k)κn−k

1∫0

(1− r2)j/2rn−k−1 dr

=12(n− k)κn−k

1∫0

(1− t)j/2tn−k−2

2 dt

=12(n− k)κn−kB

(j + 2

2,n− k

2

)=

12(n− k)κn−k

Γ( j+22 )Γ(n−k

2 )

Γ(n+j−k+22 )

=κn+j−k

κj.

Altogether this yields

αnjk =

(kj

)κkκn+j−k

Vn+j−k(Bn)κk−jκk=

(kj

)κkκn+j−k(n

k−j

)κnκj

.

This can be put in still a different form by using the identity

n!κn = 2nπn−1

2 Γ(n+ 1

2

).

46

We collect what we have obtained.

7.2 Theorem. Let K,M ∈ Kn be convex bodies and let j ∈ 0, . . . , n.Then the principal kinematic formula∫

Gn

Vj(K ∩ gM) dµ(g) =n∑

k=j

αnjkVn+j−k(K)Vk(M)

holds. For k ∈ 1, . . . , n− 1 and j ≤ k the Crofton formula∫En

k

Vj(K ∩ E) dµk(E) = αnjkVn+j−k(K)

holds. The coefficients are given by

αnjk =

(kj

)κkκn+j−k(n

k−j

)κjκn

=Γ(k+1

2 )Γ(n+j−k+12 )

Γ( j+12 )Γ(n+1

2 ).

Finally, the results are easily extended to polyconvex sets. Let K ∈ U(Kn).We choose a representation

K =m⋃

i=1

Ki

with convex bodies K1, . . . ,Km. Since Vn+j−k is additive on U(Kn), theinclusion-exclusion principle gives

Vn+j−k(K) =∑

v∈S(m)

(−1)|v|−1Vn+j−k(Kv).

Now let M ∈ Kn be a convex body. Since the principal kinematic formulaholds for convex bodies, we obtain∫

Gn

Vj(K ∩ gM) dµ(g)

=∫

Gn

Vj

(m⋃

i=1

(Ki ∩ gM)

)dµ(g)

=∫

Gn

∑v∈S(m)

(−1)|v|−1Vj(Kv ∩ gM) dµ(g)

47

=∑

v∈S(m)

(−1)|v|−1n∑

k=j

αnjkVn+j−k(Kv)Vk(M)

=n∑

k=j

αnjkVn+j−k(K)Vk(M).

Hence, the kinematic formula holds for K ∈ U(Kn) and M ∈ Kn. In asimilar way, it can now be extended to K ∈ U(Kn) and M ∈ U(Kn). Ananalogous extension is possible for the Crofton formula.

8 Extension to random sets

It has been announced in the introduction that we want to use integral-geometric results to give a theoretical foundation for some formulae usedin stereology. To achieve this goal, we shall now extend the kinematic andCrofton formulae to certain random sets.

First we have to explain what one understands by a closed random setin Rn. Let F denote the system of all closed subsets of Rn. For A ⊂ Rn

one writes

FA := F ∈ F : F ∩A 6= ∅,FA := F ∈ F : F ∩A = ∅.

The system

FG : G ⊂ Rn open ∪ FC : C ⊂ Rn compact

is a subbasis of a topology on F ; this topology is called the topology of closedconvergence. By B(F) we denote the corresponding σ-algebra of Borel sets.

Now a random closed set in Rn, briefly a RACS, is defined as a randomvariable with values in F . More precisely, a RACS is a measurable mapZ : Ω → F from some probability space (Ω,A,P) into the measurable space(F ,B(F)). For ω ∈ Ω, the closed set Z(ω) is called a realisation of Z. Theimage measure PZ := Z(P) of the probability measure P under the map Zis called the distribution of Z. Thus, this is a measure on B(F), and forA ∈ B(F) one has

PZ(A) = P(Z−1(A)) = P(ω ∈ Ω : Z(ω) ∈ A) =: P(Z ∈ A),

which is the probability that Z has a realization in the prescribed set A.The random closed set Z is called stationary if for each vector t ∈ Rn the

random closed sets Z and Z + t have the same distribution, in other words,

48

if the distribution of Z is invariant under translations. If the distributionof Z is invariant under rotations, then Z is called isotropic.

For a measurable nonnegative or P-integrable function f : Ω → R, theexpectation is

Ef :=∫Ω

f dP.

We will often have a random closed set Z : Ω → F and a measurablefunction f : F → R. If the expectation of f Z exists, it is given by

Ef(Z) :=∫Ω

f Z dP =∫F

f dPZ ,

by the transformation formula for integrals.For our envisaged applications, we have to restrict the admitted random

closed sets. The extended convex ring is defined by

LU(Kn) := F ⊂ Rn : F ∩K ∈ U(Kn) for K ∈ Kn.

The elements of LU(Kn) will also be called locally polyconvex sets. Thus alocally polyconvex set has the property that its intersection with any convexbody is a finite union of convex bodies.

If M ∈ U(Kn) is a non-empty polyconvex set, there are a number m ∈ Nand convex bodies K1, . . . ,Km ∈ Kn such that M = K1 ∪ · · · ∪Km. Thesmallest number m with this property will be denoted as N(M). We alsoput N(∅) = 0. This defines a function N : U(Kn) → N0, which can beshown to be measurable. Now we are in a position to define the randomclosed sets which will be admitted in the following.

Definition. A standard random set in Rn is a closed random set Z in Rn

with the following properties:(a) The realizations of Z are locally polyconvex,(b) Z is stationary,(c) Z satisfies the integrability condition

E2N(Z∩Cn) <∞.

Here, as before, Cn := [0, 1]n is the unit cube in Rn.

For a standard random set, one can define a volume density, a surface areadensity and, more generally, the density of the jth intrinsic volume. LetZ be a standard random set. We choose a ‘test body’ (or ‘observationwindow’) K ∈ Kn with Vn(K) > 0. For a given realization Z(ω), the

49

intersection Z(ω) ∩ K is polyconvex, hence the (additively extended) jthintrinsic volume Vj(Z(ω) ∩K) is defined. One can show that the functionω 7→ Vj(Z(ω) ∩K) is measurable, hence it defines a real random variable.Its expectation,

EVj(Z ∩K),

depends on both, the random set Z and the test body K. However, we shallsee that the limit

Vj(Z) := limr→∞

EVj(Z ∩ rK)Vn(rK)

exists and is independent of K. This number Vj(Z) is called the density ofthe jth intrinsic volume of the random set Z.

The existence proof for the limit, which is a bit technical, is precededby two lemmas. Recall that Cn is the unit cube given by

Cn := x = (x1, . . . , xn) ∈ Rn : 0 ≤ xi ≤ 1 for i = 1, . . . , n.

The set∂+Cn := x = (x1, . . . , xn) ∈ Cn : max

1≤i≤nxi = 1

is called the right upper boundary of Cn. It is polyconvex. We need the set

Cn0 := Cn \ ∂+Cn

as a ‘fundamental domain’; the space Rn can be represented as a disjointunion of translates of Cn

0 :

Rn =⋃

z∈Zn

(Cn0 + z).

We write Zn as a sequence (zi)i∈N (in any order) and put

Ci := Cn + zi, ∂+Ci := ∂+Cn + zi.

The set Cn0 belongs to U(Pn

ro), the class of all finite unions of relativelyopen convex polytopes. Below we shall use the fact that every additivefunctional on the class of polytopes has an additive extension to U(Pn

ro).We do not give a proof here, but refer to [6].

8.1 Lemma. If ϕ : U(Kn) → R is an additive function and K ∈ U(Kn) isa polyconvex set, then

ϕ(K) =∑i∈N

[ϕ(K ∩ Ci)− ϕ(K ∩ ∂+Ci)].

50

Proof. Let K ∈ U(Kn). For a polytope P ∈ Kn we define

ψ(P ) := ϕ(K ∩ P ).

Then ψ is an additive functional on convex polytopes and hence has a uniqueextension to an additive function on U(Pn

ro), also denoted by ψ. Withoutloss of generality we may assume that

K ⊂ Q :=m⋃

i=1

(Cn0 + zi)

and that Q is convex (where Q denotes the closure of Q). Then

ϕ(K) = ϕ(K ∩Q) = ψ(Q) = ψ(Q)

=∑i∈N

ψ(Cn0 + zi)

=∑i∈N

[ψ(Ci)− ψ(∂+Ci)]

=∑i∈N

[ϕ(K ∩ Ci)− ϕ(K ∩ ∂+Ci)].

Here we have used the additivity of ψ on U(Pnro) and the fact that ψ(P ) = 0

for all convex polytopes P with K ∩ P = ∅.

We call a function ϕ : U(Kn) → R conditionally bounded if, for each K ′ ∈Kn, the function ϕ is bounded on the set K ∈ Kn : K ⊂ K ′. When ϕ istranslation invariant and additive, it is sufficient for this to assume that ϕis bounded on the set K ∈ Kn : K ⊂ Cn.

8.2 Lemma. Let the function ϕ : U(Kn) → R be translation invariant,additive and conditionally bounded. Then

limr→∞

ϕ(rK)Vn(rK)

= ϕ(Cn)− ϕ(∂+Cn)

for every K ∈ Kn with Vn(K) > 0.

Proof. Let K ∈ Kn and 0 ∈ intK, without loss of generality. For z ∈ Rn

we putϕ(K, z) := ϕ(K ∩ (Cn + z))− ϕ(K ∩ (∂+Cn + z)). (42)

Lemma 8.1 shows that

ϕ(rK) =∑z∈Zn

ϕ(rK, z) for r > 0.

51

DefineZ1

r := z ∈ Zn : (Cn + z) ∩ rK 6= ∅, Cn + z 6⊂ rK

andZ2

r := z ∈ Zn : Cn + z ⊂ rK.

Then

limr→∞

|Z1r |

Vn(rK)= 0, lim

r→∞

|Z2r |

Vn(rK)= 1, (43)

where |A| denotes the number of elements of a set A. The limit relationsfollow from the fact that one easily shows the existence of numbers r0 >s, t > 0 such that

z ∈ Z1r ⇒ Cn + z ⊂ (r + s)K \ (r − s)K,

(r − t)K ⊂⋃

z∈Z2r

(Cn + z)

for r ≥ r0.By assumption,

|ϕ(rK, z)| = |ϕ(rK − z, 0)| ≤ b

with some constant b independent of z,K and r. This gives

1Vn(rK)

∣∣∣∣∣∣∑

z∈Z1r

ϕ(rK, z)

∣∣∣∣∣∣ ≤ b|Z1

r |Vn(rK)

→ 0 for r →∞.

From this we deduce

limr→∞

ϕ(rK)Vn(rK)

= limr→∞

1Vn(rK)

∑z∈Zn

ϕ(rK, z)

= limr→∞

1Vn(rK)

∑z∈Z2

r

ϕ(rK, z)

= [ϕ(Cn)− ϕ(∂+Cn)] limr→∞

|Z2r |

Vn(rK)= ϕ(Cn)− ϕ(∂+Cn).

We are now in a position to prove the existence of the densities of intrinsicvolumes for standard random sets.

52

8.3 Theorem. For a standard random set Z and for j ∈ 0, . . . , n, thelimit

Vj(Z) := limr→∞

EVj(Z ∩ rK)Vn(rK)

exists, and it satisfies

Vj(Z) = E[Vj(Z ∩ Cn)− Vj(Z ∩ ∂+Cn)].

Hence, Vj(Z) is independent of K.

Proof. Let K ∈ Kn and Vn(K) > 0. Without loss of generality, we canassume that K ⊂ Cn. For given ω ∈ Ω, there is a representation

Z(ω) ∩K =NK(ω)⋃

i=1

Ki(ω) with Ki(ω) ∈ Kn,

where NK(ω) := N(Z(ω) ∩K). By the inclusion-exclusion principle,

Vj(Z(ω) ∩K)

=NK(ω)∑

k=1

(−1)k−1∑

1≤i1<···<ik≤NK(ω)

Vj(Ki1(ω) ∩ · · · ∩Kik(ω)),

hence, by the monotoneity of the intrinsic volumes,

E|Vj(Z ∩K)| ≤ Vj(Cn)ENK∑k=1

(NK

k

)≤ Vj(Cn)E2N(Z∩K)

≤ Vj(Cn)E2N(Z∩Cn),

since N(Z(ω)∩K) ≤ N(Z(ω)∩Cn). By assumption, the right-hand side isfinite, hence Vj(Z ∩K) is integrable. For a polyconvex set M ∈ U(Kn), theintegrability of Vj(Z ∩M) then follows from additivity, using the inclusion-exclusion principle again. This shows that all expectations appearing inthe theorem exist and are finite. Therefore, we can define a functionalϕ : U(Kn) → R by

ϕ(M) := EVj(Z ∩M) for M ∈ U(Kn).

Then ϕ is additive, translation invariant (as follows from the stationarityof Z) and conditionally bounded (as follows from the last estimate above).Now the assertion of the theorem follows from Lemma 8.2.

53

After these preliminaries, we are now able to answer questions of the fol-lowing kind. Suppose that the realisations Z(ω) of a closed standard setZ can be observed in a ‘window’, that is, in a compact convex set K withVn(K) > 0.

By ‘observation’ we mean that, in principle, the values Vj(Z(ω)∩K) canbe measured. We want to use the values Vj(Z(ω) ∩K)/Vn(K) to estimatethe densities Vj(Z). But in general, Vj(Z∩K)/Vn(K) will depend on K andthus will not be an unbiased estimator for Vj(Z). To control the error, wewould have to determine the expectation of Vj(Z ∩K). If Z is an isotropicstandard random set, this can be achieved by means of integral geometry.From the obtained set of expectations, one can then also derive unbiasedestimators for the densities of the intrinsic volumes.

The next theorem extends the principal kinematic formula to isotropicstandard random sets.

8.4 Theorem. Let Z be an isotropic standard random set in Rn, let K ∈ Kn

and j ∈ 0, . . . , n. Then

EVj(Z ∩K) =n∑

k=j

αnjkVk(K)Vn+j−k(Z).

(The coefficients are those of Theorem 7.2.)

Proof. First we denote that the function

Rn × SOn × Ω → R

(x, ϑ, ω) 7→ Vj(Z(ω) ∩K ∩ (ϑBn + z))

is integrable with respect to the product measure λ⊗ν⊗P. Since E2N(Z∩Cn)

<∞, this follows as in the proof of Theorem 8.3, if we additionally assumethat K ⊂ Cn. For general K ∈ Kn it then follows from∫ ∫ ∫

|Vj(Z(ω) ∩K ∩ (Cn + z) ∩ (ϑBn + x))| dλ(x)dν(ϑ)dP(ω)

=∫ ∫ ∫

|Vj(Z(ω) ∩ (K − z) ∩ Cn ∩ (ϑBn + x− z))| dλ(x)dν(ϑ)dP(ω)

=∫ ∫ ∫

|Vj(Z(ω) ∩ (K − z) ∩ Cn ∩ (ϑBn + x))| dλ(x)dν(ϑ)dP(ω)

<∞

and the inclusion-exclusion formula.

54

For ϑ ∈ SOn, x ∈ Rn and r > 0 we deduce from the motion invarianceof Vj and the stationarity and isotropy of Z that

EVj(Z ∩K ∩ (ϑrBn + x))= EVj(ϑ−1(Z − x) ∩ ϑ−1(K − x) ∩ rBn)= EVj(Z ∩ ϑ−1(K − x) ∩ rBn).

From Fubini’s theorem (and the invariance properties of λ and ν) we get

E∫

SOn

∫Rn

Vj(Z ∩K ∩ (ϑrBn + x)) dλ(x) dν(ϑ)

= E∫

SOn

∫Rn

Vj(Z ∩ (ϑK + x) ∩ rBn) dλ(x) dν(ϑ).

We apply the principal kinematic formula (Theorem 7.2) to both sides andobtain

n∑k=j

αnjkEVk(Z ∩K)Vn+j−k(rBn) =n∑

k=j

αnjkVk(K)EVn+j−k(Z ∩ rBn).

Now we divide both sides by Vn(rBn) and let r tend to infinity. Because ofVm(rBn) = rmVm(Bn) and αnjj = 1, the left side tends to

EVj(Z ∩K)

and by Theorem 8.3, the right side tends to

n∑k=j

αnjkVk(K)Vn+j−k(Z).

This completes the proof.

The special cases

EVn(Z ∩K) = Vn(K)Vn(Z),EVn−1(Z ∩K) = Vn−1(K)Vn(Z) + Vn(K)Vn−1(Z)

of Theorem 8.4 can be obtained without the assumption of isotropy, sincecorresponding translative integral-geometric formulae are available.

Now we interpret Theorem 8.4. As one application, it describes theerror which is made if the measured value Vj(Z(ω) ∩K)/Vn(K) is used as

55

an estimator for the density Vj(Z). Writing the formula of Theorem 8.4 inthe form

EVj(Z ∩K)Vn(K)

= Vj(Z) +1

Vn(K)

n−1∑k=j

αnjkVj(K)Vn+j−k(Z),

we see that the mean error tends to 0 for increasing windows K, thus theestimator

Vj(Z(ω) ∩K)/Vn(K)

is asymptotically unbiased. However, one can also obtain an unbiased esti-mator. The system of equations given by Theorem 8.4,

EVj(Z ∩K) =n∑

k=j

αnjkVk(K)Vn+j−k(Z), j = 0, . . . , n,

can be solved for V0(Z), . . . , Vn(Z), since the coefficient matrix is triangular.This yields formulae of the form

Vi(Z) = E

(n∑

m=0

βnim(K)Vm(Z ∩K)

), i = 0, . . . , n,

hencen∑

m=0

βnim(K)Vm(Z ∩K)

is an unbiased estimator for Vi(Z). As an example, we write down the two-dimensional case, using the notations A,L, χ for area, perimeter and Eulercharacteristic, respectively:

A(Z) = EA(Z ∩K)A(K)

,

L(Z) = E[L(Z ∩K)A(K)

− L(K)A(Z ∩K)A(K)2

],

χ(Z) = E[χ(Z ∩K)A(K)

− 12π

L(K)L(Z ∩K)A(K)2

+(

12π

L(K)2

A(K)3− 1A(K)2

)A(Z ∩K)

].

Theorem 8.4 also immediately yields a Crofton formula for random sets.If we talk of a standard random set Z in some affine subspace E, the sta-tionarity and isotropy of Z refer to E, and densities of intrinsic volumeshave to be computed in E.

56

8.5 Theorem. Let Z be an isotropic standard random set in Rn, let E ∈ Enk

be a k-dimensional flat, where k ∈ 1, . . . , n − 1, and let j ∈ 0, . . . , k.Then Z ∩ E is an isotropic standard random set in E, and

Vj(Z ∩ E) = αnjkVn+j−k(Z).

Proof. We omit the (not difficult) proof that Z ∩ E is, with respect toE, again an isotropic standard random set. For that reason, the densityVj(Z ∩ E) exists. Now let K ∈ Kn, K ⊂ E and Vk(K) > 0. Theorem 8.4yields

EVj(Z ∩K) =k∑

m=j

αnjmVm(K)Vn+j−m(Z), (44)

where only terms with m ≤ k appear since Vm(K) = 0 for m > k. Since Zis stationary, we can assume that 0 ∈ E and hence rK ⊂ E for r > 0. In(44), we replace K by rK and divide the equation by Vk(rK). For r →∞,the left side tends to Vj(Z ∩ E), since Vj(Z ∩ rK) = Vj(Z ∩ E ∩ rK) (andthe intrinsic volumes do not depend on the dimension of the surroundingspace). The right side tends to αnjkVn+j−k(Z).

The implications of this theorem are clear. After Theorem 8.4, we hadseen how the densities Vj(Z) of an isotropic standard random set admitasymptotically unbiased or even unbiased estimators. If Z is observed ina k-dimensional section Z ∩ E, then we can obtain estimators for Vj(Z ∩E). Theorem 8.5 tells us that these are at the same time (asymptotically)unbiased estimators for the densities αnjkVn+j−k(Z).

As an example, we consider the practically relevant case where n = 3 andk = 2. We deal with the three-dimensional densities V (volume), S (surfacearea), M (integral of mean curvature) and with the two-dimensional densi-ties A (area), L (boundary length), χ (Euler characteristic). The equationsof Theorem 8.5 now read

V (Z) = A(Z ∩ E), (45)

S(Z) =4πL(Z ∩ E), (46)

M(Z) = 2πχ(Z ∩ E). (47)

These equations, finally, provide an exact theoretic foundation for the ‘fun-damental equations of stereology’, which are traditionally written in theform

VV = AA,

57

SV =4πLA,

MV = 2πχA.

Concluding we can say that Theorems 8.4 and 8.5 provide theoreticaljustifications for some practical procedures of stereology, at least in thosecases where it is reasonable to model probes of real materials by realisationsof isotropic standard random sets. From the practical point of view, the con-sideration of only locally polyconvex sets does not seem very restrictive. Ofthe invariance properties, stationarity is always unrealistic, requiring un-bounded sets, but it may well be satisfied approximately at close range.The most critical assumption is that of isotropy. For that reason, the ap-plicability of motion invariant stereology is limited, and translative integralgeometry is under investigation.

9 The kinematic formula for curvature mea-sures

We shall now prove the local version of the principal kinematic formula,that is, the equation ∫

Gn

Φj(K ∩ gM,A ∩ gB) dµ(g) (48)

=n∑

k=j

αnjkΦj(K,A)Φn+j−k(M,B)

for the curvature measures Φi. It holds for polyconvex sets K,M ∈ U(Kn)and Borel sets A,B ∈ B(Rn). As for the global version, involving theintrinsic volumes Vi, it is sufficient to prove (48) for convex bodies K,M ∈Kn, since the general case of polyconvex sets is then easily obtained, usingadditivity and the inclusion-exclusion principle.

For the proof of (48), we first consider the case where K and M aren-dimensional convex polytopes. We also consider only translations insteadof rigid motions, thus we have to investigate the integral

I :=∫

Rn

Φj(K ∩ (M + x), A ∩ (B + x)) dλ(x).

By (37), the jth curvature measure of a polytope P is given by

Φj(P, ·) =∑

F∈Fj(P )

γ(F, P )λF .

58

It follows that

I =∫

Rn

∑F ′∈Fj(K∩(M+x))

γ(F ′,K ∩ (M + x))λF ′(A ∩ (B + x)) dλ(x). (49)

The faces F ′ ∈ Fj(K ∩ (M + x)) are precisely the j-dimensional sets of theform F ′ = F ∩ (G+x) with a face F ∈ Fk(K) and a face G ∈ Fi(M), wherek, i ∈ j, . . . , n. In computing the integral (49), only those translationvectors x need to be considered for which a pair F,G with F ∩ (G+ x) 6= ∅also satisfies relintF ∩ relint (G + x) 6= ∅, since the remaining vectors xmake up a set of Lebesgue measure zero. Moreover, the pairs F,G forwhich k + i < n or which are in special position, do not contribute to theintegral, since for them we have

λ(x ∈ Rn : F ∩ (G+ x) 6= ∅) = λ(F +G∗) = 0.

In the remaining cases, we have dimF ′ = dimF + dimG − n and hencek + i = n+ j. Therefore, we obtain

I =n∑

k=j

∑F∈Fk(K)

∑G∈Fn+j−k(M)∫

Rn

γ(F ∩ (G+ x),K ∩ (M + x))λF∩(G+x)(A ∩ (B + x)) dλ(x).

In the integrand, we may assume that relintF ∩ relint (G + x) 6= ∅, and inthis case the external angle

γ(F ∩ (G+ x),K ∩ (M + x)) =: γ(F,G,K,M)

does not depend on x. Putting

J(F,G) :=∫

Rn

λF∩(G+x)(A ∩ (B + x)) dλ(x),

we thus have

I =n∑

k=j

∑F∈Fk(K)

∑G∈Fn+j−k(M)

γ(F,G,K,M)J(F,G).

To compute the integral J(F,G) for given faces F ∈ Fk(K) and G ∈Fn+j−k(M), we decompose the space Rn in a way adapted to these faces.We may assume that

0 ∈ L1 := aff F ∩ aff G,

59

where aff denotes the affine hull. Let

L2 := L⊥1 ∩ aff F, L3 := L⊥1 ∩ aff G,

and let λ(j), λ(k−j), λ(n−k) denote the Lebesgue measures on L1, L2, L3, re-spectively. With respect to the direct sum decomposition Rn = L1⊕L2⊕L3,every x ∈ Rn has a unique decomposition x = x1 +x2 +x3 with xi ∈ Li fori = 1, 2, 3. Writing

A′ := A ∩ F, B′ := B ∩G,

we get

J(F,G) = [F,G]∫L3

∫L2

∫L1

λF∩(G+x1+x2+x3)(A′ ∩ (B′ + x1 + x2 + x3)

dλ(j)(x1) dλ(k−j)(x2) dλ(n−k)(x3).

Here the factor [F,G] is an absolute determinant, defined by

dλ(x) = [F,G] dλ(j)(x1) dλ(k−j)(x2) dλ(n−k)(x3).

It can be described as follows, in a more general version. Let L,L′ ⊂ Rn

be two linear subspaces. We choose an orthonormal basis of L ∩ L′ andextend it to an orthonormal basis of L and also to an orthonormal basis ofL′. Let P denote the parallelepiped that is spanned by the vectors obtainedin this way. We define [L,L′] := Vn(P ). Then [L,L′] depends only on thesubspaces L and L′. If L + L′ 6= Rn, then [L,L′] = 0. We extend thisdefinition to faces F,G of polytopes by putting [F,G] := [L,L′], where Land L′ are the linear subspaces which are translates of the affine hulls of Fand G, respectively.

To compute now the inner integral over L1, we observe that

(A′ ∩ (B′ + x1 + x2 + x3))− x2 = (A′ − x2) ∩ (B′ + x1 + x3) ⊂ L1

and hence ∫L1

λF∩(G+x1+x2+x3)(A′ ∩ (B′ + x1 + x2 + x3)) dλ(j)(x1)

=∫L1

λ(j)((A′ − x2) ∩ (B′ + x3 + x1)) dλ(j)(x1)

= λ(j)((A′ − x2) ∩ L1)λ(j)((B′ + x3) ∩ L1),

60

where we have used Theorem 2.1. The integrations over L2 and L3 nowrequire only Fubini’s theorem, and we get∫

L2

λ(j)((A′ − x2) ∩ L1) dλ(k−j)(x2) = λ(j) ⊗ λ(k−j)(A′) = λF (A),

∫L3

λ(j)((B′ + x3) ∩ L1) dλ(n−k)(x3) = λ(j) ⊗ λ(n−k)(B′) = λG(B).

Together this yields

J(F,G) = [F,G]λF (A)λG(B).

Inserting this in the integral I, we end up with the following translativeintegral-geometric formula for polytopes.

9.1 Theorem. If K,M ∈ Kn are polytopes and A,B ∈ B(Rn) are Borelsets, then for j ∈ 0, . . . , n,∫

Gn

Φj(K ∩ (M + x), A ∩ (B + x)) dλ(x)

=n∑

k=j

∑F∈Fk(K)

∑G∈Fn+j−k(M)

γ(F,G,K,M)[F,G]λF (A)λG(B).

The kinematic formula at which we are aiming requires, for polytopes, thecomputation of∫

Gn

Φj(K ∩ gM,A ∩ gB) dµ(g)

=∫

SOn

∫Gn

Φj(K ∩ (ϑM + x), A ∩ (ϑB + x)) dλ(x) dν(ϑ)

=n∑

k=j

∑F∈Fk(K)

∑G∈Fn+j−k(M)

λF (A)λG(B)

∫SOn

γ(F, ϑG,K, ϑM)[F, ϑG] dν(ϑ).

Here we have used the fact that λϑG(ϑB) = λG(B). The summands with

61

k = j or k = n are easily determined, since for k = j we get∑F∈Fk(K)

∑G∈Fn+j−k(M)

γ(F,G,K,M)[F,G]λF ⊗ λG

=∑

F∈Fj(K)

γ(F,M,K,M)[F,M ]λF ⊗ λM

=∑

F∈Fj(K)

γ(F,K)λF ⊗ λM

= Φj(K, ·)⊗ Φn(M, ·),

and similarly for k = n,∑F∈Fk(K)

∑G∈Fn+j−k(M)

γ(F,G,K,M)[F,G]λF ⊗ λG

= Φn(K, ·)⊗ Φj(M, ·).

The remaining integrals over the rotation group are determined in thefollowing theorem.

9.2 Theorem. Let K,M ∈ Kn be polytopes, let j ∈ 0, . . . , n − 2, k ∈j + 1, . . . , n− 1, F ∈ Fk(K) and G ∈ Fn+j−k(M). Then∫

SOn

γ(F, ϑG,K, ϑM)[F, ϑG] dν(ϑ) = αnjkγ(F,K)γ(G,M),

where αnjk is as in Theorem 7.2.

Proof. In order to avoid difficult direct computations, we will give a proofbased on the uniqueness of spherical Lebesgue measures. This is possiblesince external angles are defined in terms of such measures.

By definition,

γ(F, ϑG,K, ϑM) = γ(F ∩ (ϑG+ x),K ∩ (ϑM + x))

with suitable x ∈ Rn. As before, let N(P, F ) denote the normal cone of apolytope P in a relatively interior point of its face F . From the definitionof the external angle we get

γ(F, ϑG,K, ϑM) =σ(L)(N(K ∩ (ϑM + x), F ∩ (ϑG+ x)) ∩ Sn−1)

σ(L)(L ∩ Sn−1),

where L ∈ Lnn−j is the orthogonal space of F ∩(ϑG+x) (i.e., the orthogonal

complement of the linear subspace parallel to the affine hull of F ∩ (ϑG +

62

x)). For a linear subspace L ⊂ Rn, we have denoted by σ(L) the sphericalLebesgue measure on L ∩ Sn−1.

A general property of normal cones of convex bodies gives

N(K ∩ (ϑM + x), F ∩ (ϑG+ x)) = N(K,F ) + ϑN(M,G).

Therefore, we have to evaluate the integral∫SOn

σ(L1+ϑL2)((N(K,F ) + ϑN(M,G)) ∩ Sn−1)[F, ϑG] dν(ϑ),

where L1 is the orthogonal space of F and L2 is the orthogonal space of G.More generally, we define the integral

I(A,B) :=∫

SOn

σ(L1+ϑL2)(C(A) + ϑC(B)) ∩ Sn−1)[F, ϑG] dν(ϑ)

for arbitrary Borel sets A ⊂ L1 ⊂ Sn−1 and B ⊂ L2 ⊂ Sn−1, where

C(A) := αx : x ∈ A, α ≥ 0

denotes the cone spanned by A. Concerning the measurability of the inte-grand, we give the following hints for a proof. The function ϑ 7→ [F, ϑG] iscontinuous, hence measurable. Let U denote the set of all rotations ϑ ∈ SOn

for which L1 and ϑL2 are not in special position. Then it can be shownthat ν(SOn \ U) = 0. For ϑ ∈ U we have

dimL1 + dimL2 = (n− k) + (k − j) = n− j ≤ n,

hence the sum L1 + ϑL2 is direct. From this one can deduce that C(A) +ϑC(B) is a Borel set (in general, the sum of two Borel sets need not bea Borel set). For different ϑ ∈ U , the sets C(A) + ϑC(B) are connectedby linear transformations. All this together is sufficient to show that themapping

ϑ 7→ σ(L1+ϑL2)((C(A) + ϑC(B)) ∩ Sn−1)

is measurable on U .For fixed B ∈ B(L2 ∩ Sn−1) we now define

ϕ(A) := I(A,B) for A ∈ B(L1 ∩ Sn−1).

If⋃∞

i=1Ai is a disjoint union of sets Ai ∈ B(L1 ∩ Sn−1), then(C

( ∞⋃i=1

Ai

)+ ϑC(B)

)∩ Sn−1 =

∞⋃i=1

((C(Ai) + ϑC(B)) ∩ Sn−1)

63

for ϑ ∈ U , and this union is disjoint up to a set of σ(L1+ϑL2)-measure zero.We deduce that

σ(L1+ϑL2)

((C

( ∞⋃i=1

Ai

)+ ϑC(B)

)∩ Sn−1

)

=∞∑

i=1

σ(L1+ϑL2)((C(Ai) + ϑC(B)) ∩ Sn−1)

for ϑ ∈ U and thus

ϕ

( ∞⋃i=1

Ai

)=

∞∑i=1

ϕ(Ai),

by the theorem of monotone convergence. It follows that ϕ is a finite mea-sure on L1 ∩ Sn−1. Let ρ ∈ SOn(L1) be a rotation mapping the subspaceL1 into itself. Then

C(ρA) + ϑC(B) = ρ(C(A) + ρ−1ϑC(B))

and[F, ϑG] = [ρF, ϑG] = [F, ρ−1ϑG],

hence

ϕ(ρA)

=∫

SOn

σ(L1+ϑL2)((C(ρA) + ϑC(B)) ∩ Sn−1)[F, ϑG] dν(ϑ)

=∫

SOn

σ(L1+ρ−1ϑL2)((C(A) + ρ−1ϑC(B)) ∩ Sn−1)[F, ρ−1ϑG] dν(ϑ)

= ϕ(A).

Since spherical Lebesgue measure is uniquely determined, up to a factor, byits rotation invariance (and finiteness), the measure ϕ must be a constantmultiple of σ(L1). Analogously we deduce that for fixed A ∈ B(L1 ∩ Sn−1)the measure I(A, ·) must be a constant multiple of σ(L2). Both resultstogether yield that

I(A,B) = α(L1, L2)σ(L1)(A)σ(L2)(B)

for all A ∈ B(L1 ∩ Sn−1), B ∈ B(L2 ∩ Sn−1); here α(L1, L2) is a constantdepending only on L1 and L2. If we choose A = L1 ∩Sn−1, B = L2 ∩Sn−1

and observe the invariance properties of the functional I following from its

64

definition, we see that α(L1, L2) depends only on the dimensions n, j, k.Therefore, there is a constant βnjk so that

I(A,B) = βnjkσ(L1)(A)σ(L2)(B).

In particular, this shows that

I(N(K,F ) ∩ Sn−1, N(M,G) ∩ Sn−1)) = βnjkγ(F,K)γ(G,M).

This is the assertion of Theorem 9.2, except that it remains to show thatβnjk = αnjk.

Collecting the results obtained so far, we have proved the following kine-matic formula for polytopes K,M ∈ Kn:∫

Gn

Φj(K ∩ gM,A ∩ gB) dµ(g)

=n∑

k=j

βnjk

∑F∈Fk(K)

∑G∈Fn+j−k(M)

γ(F,K)γ(G,M)λF (A)λG(B)

=n∑

k=j

βnjkΦk(K,A)Φn+j−k(M,B).

If we choose A = B = Rn, the obtained formula must coincide with thatof Theorem 7.2, for all polytopes K,M . This shows that βnjk = αnjk andthus completes the proof of Theorem 9.2.

For arbitrary convex bodies K,M , the general kinematic formula (48) isnow obtained by approximation, using the weak continuity of the curva-ture measures. An extension to polyconvex sets K,M is easily achieved byadditivity, as in the case of Theorem 7.2.

Also the Crofton formula of Theorem 7.2 has a local counterpart. Wecollect both results in the following theorem.

9.3 Theorem. Let K,M ⊂ UKn be polyconvex sets, let j ∈ 0, . . . , n, andlet A,B ∈ B(Rn) be Borel sets. Then the principal kinematic formula∫

Gn

Φj(K ∩ gM,A ∩ gB) dµ(G) (50)

=n∑

k=j

αnjkΦj(K,A)Φn+j−k(M,B)

65

holds. For k ∈ 1, . . . , n− 1 and j ≤ k the Crofton formula∫En

k

Φj(K ∩ E,A ∩ E) dµk(E) = αnjkΦn+j−k(K,A) (51)

holds. In both cases, the coefficients αnjk are those given in Theorem 7.2.

Proof. It remains to prove formula (51). Here we can assume that K isa convex body, since the general case is then obtained by additivity. Wededuce (51) from (50), by a similar but simpler argument as used in theproof of Theorem 7.1.

Let Lk ∈ Lnk be a fixed subspace; then µk = γk(λ(n−k)⊗ν), as in Section

3. Let W be a unit cube in Lk. Let A ∈ B(Rn). By (50) we have

J :=∫

Gn

Φj(Lk ∩ gK,W ∩ gA) dµ(g)

=n∑

m=j

αnjmΦm(Lk,W )Φn+j−m(K,A)

with

Φm(Lk,W ) =

λLk

(W ) = 1 for m = k,

0 for m 6= k,

henceJ = αnjkΦn+j−k(K,A).

On the other hand,

J =∫

SOn

∫Rn

Φj(Lk ∩ (ϑK + x),W ∩ (ϑA+ x)) dλ(x) dν(ϑ)

=∫

SOn

∫L⊥k

∫Lk

Φj(Lk ∩ (ϑK + x1 + x2),W ∩ (ϑA+ x1 + x2))

dλ(k)(x2) dλ(n−k)(x1) dν(ϑ).

For the computation of the inner integral, we put

Φj(Lk ∩ (ϑK + x1), ·) =: ϕ, ϑA+ x1 =: A′.

Then ∫Lk

Φj(Lk ∩ (ϑK + x1 + x2),W ∩ (ϑA+ x1 + x2)) dλ(k)(x2)

66

=∫Lk

ϕ((W − x2) ∩A′) dλ(k)(x2)

= ϕ(A′)λ(k)(W )= Φj(Lk ∩ (ϑK + x1), Lk ∩ (ϑA+ x1)),

where Theorem 2.1 was used. This yields

J =∫

SOn

∫L⊥k

Φj(Lk ∩ (ϑK + x1), Lk ∩ (ϑA+ x1)) dλ(n−k)(x1) dν(ϑ)

=∫

SOn

∫L⊥k

Φj(K ∩ ϑ(Lk + x), A ∩ ϑ(Lk + x)) dλ(n−k)(x) dν(ϑ)

=∫En

k

Φj(K ∩ E,A ∩ E) dµk(E),

where we have used the rigid motion covariance of the curvature measuresas well as the inversion invariance of the measures λ(n−k) and ν. The tworepresentations obtained for J together prove the assertion.

References

[1] H. Groemer, On the extension of additive functionals on classes ofconvex sets. Pacific J. Math. 75 (1978), 397 – 410.

[2] H. Hadwiger, Vorlesungen uber Inhalt, Oberflache und Isoperimetrie.Springer, Berlin 1957.

[3] D.A. Klain, A short proof of Hadwiger’s characterization theorem.Mathematika 42 (1995), 329 – 339.

[4] D.A. Klain and G.-C. Rota, Introduction to Geometric Probability.Cambridge University Press, Cambridge 1997.

[5] G.-C. Rota, Geometric probability. Math. Intelligencer 20 (1998), 11 –16.

[6] R. Schneider, Equidecomposable polyhedra. In: Colloquia Math. Soc.Janos Bolyai 48 (Intuitive Geometry, Siofok 1985). North–HollandPubl. Co., Amsterdam–Oxford–New York 1987, pp. 481 - 501.

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[7] R. Schneider, Convex Bodies: the Brunn-Minkowski Theory. Cam-bridge University Press, Cambridge 1993.

[8] R. Schneider and W. Weil, Integralgeometrie. Teubner, Stuttgart 1992.

[9] R. Schneider and W. Weil, Stochastische Geometrie (in preparation).

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